Bonded Interactions and the Crystal Chemistry of Minerals: a Review # Z

This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 1 ), c –O –O r –O, ( r ) ratio c r ( G = j ) c –O bonds are r , ( IV V –O bond and one j ) in the internuclear c r ( –O bonds are intermedi- V –O and Mg– , Kevin M. Rosso –O and S– II –O, Na– ), on a one-to-one basis, indicating c r ( ) dominating ) ratio increases for each of the M– r c –O, P– c r r ( ( –Si bonded interactions in coesite is indi- G r )/ c –O– r –O, B– ( –O bonds is shared covalent and that the Be– G –Si angles. In the case of the nonbridging Si– , Charles T. Prewitt I Notwithstanding its origin in Pauling’s electrostatic The shapes and arrangements of the bond and lone pair The –O, Si– –O bonded interactions agrees with the value of the –O– region, typical of an ionic bond. bond strength rule,Si– the Brown-Shannon bondelectron valence density, for bonds along separateand trends the with coordinationthat decreasing number the bond ratio of length istion the a of M measure the of atom, interactions bond suggesting character. in An terms examina- of the the greater the bondtion. strength, Mappings the of moredensity the shared Laplacian, distribution the the and interac- for deformation the several electron electron silicates arespherical localization reviewed. domains function The ascribed mapsthe display to bond hemi- bond vectorscribed pair and to electrons larger loneSi– along kidney-shaped pair electrons domainsbonded on as- interactions, the theroom reflex shaped O domains. sides atomsagree With of in are few number the exceptions, cappedVSEPR and the by location model domains with mush- fortive those closed-shell sites embodied molecules, in ofprotonation. the defining potential The reac- electrophilicity electrophilicing of attack the the and Si– Ocated centers atoms to of compris- increasefor with understanding decreasing the angle, protonization providing of the a structure. basis features displayed byand the the bridging nonbridging Oon O atoms atoms an in in one-to-one quartzcates forsterite and that basis are possess to transferable both bridging sheet and and nonbridging O chain atoms. magnesiosili- indicates that theclosed shell Li– ionic interactions,of that the the S– C– Al– character of thetions. intermediate In andcreases contrast, shared with bonded decreasing the bondactions interac- local length with for kinetic closed shell energy inter- density in- that the Pauling bond strength is a direct measure of þ ) c r ( G , Nancy L. Ross III , each in- ¼ c ) r c r ( H –O bond lengths, VI ¨nchen disilicic acid molecule, , David F. Cox 10.1524/zkri.2008.0002 7 II O 2 ), the local potential energy, –O bond lengths and the c DOI Si r 6 ( G –O bond energy and the bond criti- and Armin Kirfel V (2008) 1–40 / , Robert T. Downs ,I 223 Connections established during last century * –Si angle, the Si– ), and the), electronic evaluated energy at density, the bond critical points, c c * Correspondence author (e-mail: [email protected]) by Oldenbourg Wissenschaftsverlag, Mu –O– r r Mineralogisch Petrologisches Institut, Universitact Bonn, Poppelsdorfer Schloss, 53115, Bonn, Germany Chemical Science Division, and W.R. Wiley Molecular Sciences Laboratory, Pacific Northwest Laboratory, Richland, Washington, USA Department of Geosciences, Virginia Tech, Blacksburg, VA 24061, USA Department of Chemical Engineering, Virginia Tech, Blacksburg, Va. 24061, USA Department of Geosciences, University of Arizona, Tucson, AZ, 85721, USA GKSS, Max-Planck-Strasse, 21502, Geesthacht, Germany ( ( V Thomas Lippmann Abstract. between bond length,and radii, crystal bond andfollowed strength, molecular by bond chemistry aelectron valence are density briefly survey distributions reviewed of forrepresentative molecules, a the recently variety generated physical ofciples with minerals local properties first-prin- and energy ofThe density structures the quantum for mechanicalzero several methods. minerals, pressure geometry-optimized andagree at at with a variety thecent. of experimental pressures structures The were within found experimental a to few Si– per- Bond critical point /Local Silicates energy / densities Sulfides /Electron / Eelectron lone density pair / domains BondElectrophilicity / strength Molecular / chemistry / Si– cal point properties forthose crystal calculated quartz for are comparable the with H VI II III IV V G. V. Gibbs I Bonded interactions and the crystal chemistry of minerals: a review # Z. Kristallogr. Received May 7, 2007; accepted August 15, 2007 an indication thatgely the bonded short interactions rangedmodel in silica experimental and are electron localand lar- density in second distributions row nature. forwith metal The first M high topology atoms resolutioncrystal bonded of and X-ray to diffraction high O, dataogy energy determined are of compared synchrotron theoretical with distributionsples single the calculated topol- methods. with firstaccumulated As princi- between the pairstions of electron show bonded that density atoms, thebond the nuclei is lengths are distribu- and progressively progressively Concomitant shielded the as with bonded the the radii decreasethe of in local the the kinetic atoms M– energy, decrease. V creases in magnitude withnating the the local kinetic potential energyfor energy density domi- intermediate in the andbonds, internuclear shared the region more interactions.sity, negative The the the greater shorter local the the electronic stabilization energy and the den- greater the shared This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. et al. cen- th G. V. Gibbs, R. T. Downs, D. F. Cox The topological and physical properties of the electron likewise be determinedin in the a modeling moreactions. and accurate study The way of electron and“extraordinary” the density used importance individual distribution in bonded isall inter- that a of it property the containsstate of information in material that principle like canincluding a be the mineral known kinetic, ordeed, about potential a studies a of representative and ground bond molecule, totaltions lengths energies and have electron [3]. played densitybonding In- distribu- theory a and10]. key the Although crystal role these studies chemistry haveknowledge in of done and much the minerals to understanding [4– advanceour development our of grasp of the of the bondedties a crystal of interactions, chemistry minerals and isstep the still physical in far proper- the fromchemistry, Bader advancement complete. and In of his anand our widely colleagues important used understanding [2] theory forged of during a the crystal powerful last part of the 20 tury forbonded classifying, interactions characterizing for aing and wide minerals in variety terms determining of ofties materials, the the topological of includ- and their physical proper- electrondensity properties density and distributions, the bond the length local variations. energy density distributions determined sincedecade the for latter a partcules of variety will last of be mineralsdegree explored and to which in representative the mole- thisdensity properties distributions review of agree the inprinciples experimental with terms electron quantum those of mechanical calculated (1)correspondence chemical with the methods, first between (2)mineral the the and the structural physicaldistribution, properties (3) properties of the theand of connection electron potential density between energy a as properties the embodied of local in the kinetic the the bonded stabilization local interactions of virialto the theorem, which bonded bond bond interactions, lengths lengthwith (4) and first and angles the principles of extent experimentally, methods a (5) agree mineral whether determined a withsity mapping those of distribution determined the and electroncapable den- the of electron locatingand localization protonization sites and function ofatoms, prediction is potential of (6) the electrophilic whetherelectron positions attack the density of distribution the bond calculatedlike H lengths for quartz a and silicate agree anglesmolecules crystal with and and those finally the of (7) calculated minerals whether for can the be representative erties bonded classified and in interactions terms thetron of local the density energy physical distribution. density prop- tives The properties extent of have to the whichhowever, as been these elec- an objec- introduction, metyet some important will of connectionsthe the that be years earlier have empirical between been recounted.and established bond bond over length, valence Beforehand, will ionic be radii, briefly bond discussed. strength Connections between structure, bondand length ionic radii Prior to theknown beginning about of the the structuresminerals twentieth and other the century, little bonded than interactions was of the brilliant speculations by such ) – c r Ri- ( –Ni r ) and 2 c –S and r r ( r ) and –S, Ni– c r ( r –S data scatter along –S bond length being ) ratio indicates that the Fe– c r ( are in close agreement with those calculated. G = j 2, ) S c –S bond length. The properties of the Ni– 3 r ) for a given low spin Fe– ( –O bonds, with the experimental bond lengths, the c V r j ( If our understanding of minerals in their natural envir- Bond critical point properties calculated for Ni bearing The successful reproduction of the bond lengths and an- –Ni bonded interactions are intermediate in character. r 2 The values. The highparallel and but low separate spin trends Fe– with the values of larger than thosespin calculated Fe– forbonded a given interactions comparablesulfides calculated high are and virtually theNi observed same metal. for No as bond thoseof the paths calculated the were Ni for found face bulk betweenbond sharing the critical octahedra Fe point of atoms propertiesdite, troilite. for Ni The the experimental Ni sulfide heazlewoo- ‘If in some cataclysm,stroyed, all and of only scientific onegeneration knowledge of sentence were creatures, was de- whatmost passed statement on information would to in containatomic the the the next hypothesis fewest that words?little I particles all believe that thingstracting it move are each is around other made in the but when of perpetual repelling they motion, atoms upon are at- being a little squeezed distance into apart, one another.’ Introduction onments together with theirproperties crystal and chemistry their andglasses, physical manifold molecular sieves, uses ceramics,devices in catalysts are the and electrical manufacture toworking of be knowledge improved,atomic of then level the it beproperties is realized. bonded are important to This interactionsimplemented. be that is Of fully at a particularly exploited thesome the true and way physical their on if propertiesbe the uses the bonded uniquely further that interactions, identified dependtion. only with Among in a an these, handful bond individualit can length bonded provides is interac- an specialthe in unique greater the measure the sense of accumulation that binding the of region, strength the the[2]. electron of shorter density a With and in bond, the theintensity the stronger recent a invention synchrotron givensources of bond with short single discriminating wavelength, arearate crystal data detectors, high can relatively now accu- physical X-ray be property, determined diffraction and the an even electron more density robust distribution, can chard P. Feynman [1] shorter the bond lengths, the greater the sulfides and high andcussed. low The spin Fe propertiesthe bearing correlate M– sulfides are linearly, dis- as observed for ate in character. Itsely is parallels noteworthy Pauling’s that classificationnegativity the differences based classification between on clo- the the M electro- and O atoms. 2 r Ni– gles for severalelectron silicates, the density comparable distributionspotential properties and chemical of reactivity the the recountedwell location in for of the the review sitesthe exploitation bodes of of deciphering the ofprinciples properties computational crystal of quantum chemical chemical minerals strategies. and problems, using first This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , , 3 2 3 A ,as 4 (OH) 2 ) SiO 2 11 O 4 (Si 7 –O bond length as , a task that would have been a ”, a result that was considered 2 A , anthophyllite, Mg 2 (OH) 2 ) –O separation for an ideal silicate tetra- 5 regions delineated by weak X-ray struc- for the oxide anion “leads directly to an O (OH) 2 2 ) A i.e. (Si 11 3 O 4 (Si times the O– 5 –O distance of 1.62 1/2 Mg ) With the ionic radii, the structures of a host of silicates With the determination and refinement of many struc- 2 –O bond were addressed together with the great diffi- 8 Si– = 3 model. The complexitiesSi– of the covalentculties character encountered ofprinciples in the deriving methods. bondmuch He lengths ‘simpler’ observed, with to( however, first- determine that the it Si– was hedral oxyanion. Thus,radius an of ionic 1.32 model that uses an ionic to be a testamentschmidt’s of radius the of accuracyconsistent the of wave the anion. functions Wasasjerna-Gold- dent, published Given later the however, [21], relativisticionic that it self- radii was the and evi- thethe Wasasjerna-Goldschmidt-Pauling Bragg’s radii atomic calculated radii foroutermost [17] the valence correlate electron with shells densitying maxima of the for the radii the with atoms a of theoretical interest, underpinning. and provid- related minerals wereblages, pictured consisting as of closelocated large packed anions in assem- with theordinated the available smaller voids octahedrally cations and, and tetrahedrally in co- some cases like calcite, CaCO IV and talc, Mg with the Cnated atom voids evenly [4]. located It in[22] was the proposed during triangularly his this coordi- served famous time-frame to that set revolutionize Pauling ofbonded the structural interactions way postulates and thatof that rationalized mineralogists minerals. the viewed He crystaladditive regarded chemistry the sum bond ofvalence length electrons the as of ionic given theuted by radii metal the among atoms and tonumber the later be of considered bonds equally which distrib- relative the of was sizes the of assertedasserted the coordinating to that cations be anions, the andanion sum determined the the in of by anions a thethe the stable [23]. bond valence structure He strengths requirements would reaching also tant exactly of each or constraint each nearly anion. onionic satisfy With radii the this and bonded impor- ing radius interactions of ratio a together considerations silicate and with structure the such pictur- as forsterite, Mg ture amplitudes, Bragg26] and were his colleagues ablevery [4, to 13, complicated solve 14,Ca 25, the silicates crystal structures like of several diopside, tremolite, consisting of anions assemblage with of theoctahedral tightly smaller and Mg packed tetrahedrally and coordinatedsible oxide voids, Si to an- it cations account wasstructures in for pos- [24–26]. the a Also, available for large by variety assuming the of astraints oxide the radius of imposed known anion, 1.35 crystalgroup on by symmetry and a taking considerationsbidden guided periodic advantage to by Si, of structure regions for- the by con- the space challenge to solve eventhe with 1960s. the strategies Clearly,knowledge available ionic during of radii diffraction used andling’s in space postulates conjunction group played with theory amuch and a fundamental of Pau- role theduring in the establishing structural early part foundations of the of last crystaltures century. leading chemistry up to the 1970s, a wealth of relatively accu- –O denotes a Si– IV , by Sir Lawrence Bragg [16] for the oxide anion. The close 6 O 2 A . During the same time frame, Pauling [19] 1 for the oxide anion, the famous geochemist Gold- A For sake of convenience and clarity, the coordination number of From the bond lengths, numerous sets of ionic radii 1 –O bonded interaction involving a four-coordinated Si atom. schmidt [15] used Bragg’sprehensive [17] set strategy to of generatewere ionic a in radii com- popular that useWith bear the until his the radii, name, latter rulesdination radii numbers quarter were that for of devised the last forsiderations cations century. predicting based the on radius coor- ratio con- agreement between the Paulingtaken and as Goldschmidt radii aradii was confirmation were that correctcates and the that are Wasasjerna-Goldschmidt the largelyechoed bonded ionic in interactions a in in latertype sili- character. study of for This the conclusion MgAlwho physical properties was concluded oxides and that bond andunderstood the silicates in properties terms by are much of Verhoogen more [20] an easily ionic rather than a covalent completed a theoreticalcrystals study of and the derivedusing sizes quantum of a ions mechanically setthe in derived Born-Lande ionic of lattice screening energy semi-empiricalbond constants, model lengths ionic and to a radii scaleionic set the of radius radii observed that of resulted in 1.40 an effective an atom will besuperscript denoted to by the a symbol Roman of numeral the attached atom; as for a example, preceding were derived, utilizing athat strategy assumed pioneered that bygarded the Bragg bond as [17] lengthsatoms. the of Using crystals additive Wasasjerna’ssurements can sums [18] be and careful re- of1.32 his refractivity the mea- determination radii of of a spherical fixed radius of were available for studyingcomputational bonded interactions. difficulties Giventhese the maps, encountered workersprimarily on in at solving the crystalmodern determining structures. time methods Despite focused thebond for lack their lengths of determining attention determinedaccurate. crystal in The accuracy structures, the wasthe studies the made bulk possible were of by thematerials remarkably the structures like fact determined rock that werecial salt high position where symmetry and eachfrom the a atom bond knowledge occupies lengths of are a the unit determined spe- cell directly dimensions [4, 15, 17]. workers as Barlow [11,rock 12] salt who crystal, pictured forclosely the structure example, packed of as spherical a aarray Na cubic of and periodic molecules, Cl arraythe notwithstanding of atoms the time rather prevailing that viewdiscovery than of at all an X-rays crystals indiffraction 1895 consist methods and of and the equipment(cf. development molecules. of during [13]), With X-ray the not the largely early only correct, 1900s but were theber Barlow’s structures of of pictures a minerals proven relativelywas were large to made determined num- be in andproperties, substantial forging headway an structuralbonded understanding interactions of relationships, minerals, [4,perimental their 14, difficulties classification 15]. andthe But, the measurement because and hard ofand of work accurate the the encountered ex- absence X-raydensity in maps of diffraction such computers datafor as at sets the diopside, elegant the ones CaMgSi time, crafted few at that electron time Bonded interactions and crystal chemistry: a review Si– This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. et al. –O), the (M– (O), reaching R z –O bond lengths for (O) departing from 2.0 by z , and the M– s (O) correlates with (O), typically the shorter the bond G. V. Gibbs, R. T. Downs, D. F. Cox z z bond lengths. After all, it was the bond lengths (valence of a metal atom/coordination number). i –O) n Despite the successful use the Shannon-Prewitt [29] –O bonded atoms is largely independent of the forces (M– R each of the oxidethe anions magnitude in aoxide of structure anion. the tends But, to asdeparts ionic equal observed from valence 2.0, later 2.0 byside. requirement for Bragg Later several of [4], of the asfined, the the sum the more oxide rule structureslarge anions was number in were of found diop- minerals determined to with and be re- violated for a relatively and the Shannon [30]mical radii properties in correlating and physicalture in and field che- maps, the O’Keeffethe fruitful and radii construction Hyde are [39, ofpredicting little 40] struc- the more observed bond that than lengthscluded a in set that crystals. of the Asare contact such, Shannon distances they in and for con- h Prewitt reality table a of table oxide radii of Shannon and Prewitt average as much as 40%later [44]. by As firstsmaller Baur reported the by [44], value Smith [45] of and Connections between bond length and bond valence As observed above,was the elevated crystal to aling chemistry much [22] of more the proposedserved exploitable silicates his that level when a famous Pau- ite number set are of of structures in postulatesstates like exact quartz that and conformity and the with ob- forster- sum his of second postulate the that bond strengths, that were determinednot experimentally the in radii. derivingatom As the in observed radii, a by moleculeunobservable. Cremer (or According, and a crystal) Krakaradius it is [41], of a necessarily the an quantum atombond follows mechanical is likewise length that an is the length, unobservable whereas not the the the observable.considered radii when As of deciding such, thein whether it one a bonded bonded structure atoms isstructure interaction can that the is be should favored bond replaced over be structural another by field based another map. on orthat its After whether is position all, one connected ininteraction, it a to not is the the the radii energyto of length and predict the of stability bonded the ainteractions, of atoms bond by that a bond treating are length bonded radii used it [42]. as implies In fixed and termsM– that strictly of additive, the theexerted contact bonded on distance theby coordination between the polyhedron other aconstruction housing parts of pair the of comprehensive of sets pair and the of structure their radii [43]. use (bondand Nonetheless, lengths) in the the modeling thesubstantial of construction physical improvements of properties inchemistry. has structure-field our clearly maps understanding led to of crystal lengths. With thegues observations [46, by 47] Paulingand that and related his the materials can collea- bondber, be lengths ranked in inIn terms molecules, a of metals subsequent bond application num- and for Shannon the [50] oxides showedempirical [48, that bond 49], the valence, Brown connection between the ,on A 0.01 , for oxides coordinated [19]. These r –O), was re- n A (M– R -fold MO n –O bond lengths, plots and assuming that the six-coordinated 3 r –O bond lengths and the unit cell dimensions for –O bond lengths for given In his 1985 Presidential address to the Mineralogical So- ported. This information providedclassification a basis of for the mineralsthem systematic and and appreciating for their[7, discriminating similarities 27, and between 28]). differences With1000 (cf. the carefully bond determined lengthsPrewitt crystal provided [29] by structures, more Shannon and than precise and later sets of Shannon effective [30]derived ionic and to determined crystal date, the radii, ume using most versus Bragg’s strategy [17], unit cell vol- rate metal-oxygen M– 4 fixed ionic radius of the oxide ion is 1.40 ciety of America ona the ‘qualitative evolution to of a crystal quantitativethat chemistry endeavor’, the from Prewitt two [37] most stated ern important the determinative crystal factors chemistry ofthe that a nominal gov- mineral oxidation are states thetion ionic of to radii the reproducing and the bondedof bond atoms. substantial lengths, In use the addi- in radiichemistry improving proved the when to understanding be used ofdiagrams in crystal that the map construction theture relative of structural in structure-field stability terms of ofand a crystal a struc- radii. range These of maps,ler as chemical ably and substituents demonstrated and by Royrange Mul- ionic [38], of ionic provide radiiset a of and crystal basis chemical structures substituentssubstantial can for tolerate. use that understanding Further, in a the the the given radiidiffusion, modeling site found of preferences, ion surface conductivity, tension,size defects, chemical discrimination zoning, anding leaching, together of with tracecoexisting the elements phases. model- The and radiiworkers distribution have as coefficients also guides in been among thesubstitution used synthesis by of of earlier new an materialssimilar with atom radius the in andture. a oxidation state structure As to for emphasizedsynthesis produce another by of a with new Prewitt newthe materials a struc- [37], ionic resulted the from radiiions success largely and for matching in the structures of the oxidationof interest, high states for pressure silicate example, of analogueswidely in the of the germanates, substituent synthesis used a strategy changes in in corresponding silicates. predicting pressure induced structural polyhedra for a widecules variety are of different roughly crystals thestructure and type same [29, mole- and 30]. largely independent of the average, when the coordination numbers,and nominal electronic oxidation spin statescount. of The main the reason atomsso why were successful the taken in strategy into reproducing ofracy ac- fixed bond can radii lengths be was ascribed withage to such M– the accu- experimental fact that the aver- highly cited radii have sincethey proven take to be into of account greattion the use coordination states in numbers that and and(when the oxida- relevant) electron together spinthe with states oxide of the anion [31]. the coordinationthat Following metal number the the derivation, M– atoms of it was found a large number ofthe earth materials radii are on highlylarger correlated an the with cell one-to-one edges basis,36]). and the Further, the unit the larger cell averagepolyhedra volumes the bond (cf. were lengths radii, [28, for reproduced 32– the the typically coordinated within This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. s 5 A b b , cal- , the r s / (MO) s R is the row r , and the row s where r / ) for the same bonded s b –O data are plotted as open 40 oxide crystals, scatter along quasi-parallel bond lengths were plotted s , of the M atom. Figure 1a a r i . Indeed, when the bulk of the and –O) 0.22 ) the Brown-Shannon [50] empirical bond i ) a . r (M– / the bond lengths for all six rows of the s R 0.22 –O) h , the data scattered along a single trend data set likewise scattered along a single r, r ) / –O), again two trends emerged [54], one s r / (M– s/r a 1.39( s R (M– h R –O bonded interactions involving first row and 1.39( i¼ , of the bonded interactions and ( ¼ s Bond lengths observed for –O) –O) vs. –O) (M– (M– (M– R was divided by (Fig. 1b)h modeled by the power law expression nonlinear power law-trends,the other one second involving row first periodic table row M and atoms. But, when for the Mof and O neutral atoms silicatenected and molecules those with [53] calculated the were for a found Pauling variety to bond be strength, con- periodic table were foundtrend to as scatter establishedwhen along the the for Brown-Shannon same the empiricalculated single molecules bond for valences, [53, arespect 54]. large to Further, number of silicates were plotted with with respect to symbols and thesion second row analysis areR plotted of as solid the symbols. A data regres- in (b) also yielded the expression number of theupper M right atom. insert The of symbols (b). The for first the row bonds M– are given in the plotted with respect to ( number of periodicshows table, that R for the M– the other for secondempirical row M bond atoms strengthsnumber, (Fig. 2a). were and But, when each the the divided bond by lengths the row plotted against Shannon-Prewitt interactions plotted with respect to the ratio valence, Fig. 2. , N i ) can R .As –O) N A (M– –O)/ and R h 0.01 R (M– R ( ¼ s –O) and the bond va- –O bond lengths corre- (M– R –O and s where –O bond lengths, r / s , (obtained –O interac- A [53]. i 0.22 –O) ) s/r (M– R h –O bonded interaction and where 1.39( was defined to be the empirical bond valence for ¼ s Shannon-Prewitt [29] average ) yielded the expression b –O) Molecular orbital geometry-optimized bond lengths for Earlier, the average M– is the row number of the M metal ) with respect to the ratio ) Pauling’s [22] bond strength, (M– b a bonded interactions anddenote the the second red row ones M– r atom in thespheres periodic denote table. the The blue first rowtions. M– A regression analysisin of the data R ( determined by adding the Shannon and Prewitt [29] radii a numberfirst of and neutral secondPauling’s row hydroxyacid bond M strengths atoms molecules for have the containing demonstrated M– that the be viewedside as constraint Lagrangian thateach multipliers of the obtained the sumnominal with M of bond the and valence bondbond O lengths, of valences atoms the the reaching pression sums in were atoms. of a found valences With to structurestructures obtained match the to matches with the within observed valences the thening a for ex- few for a percent, the varietybond providing of existence length an of and underpin- a bond valence. close correspondence between late linearly withsuggesting the a Mulliken bond connectionthe overlap between accumulation populations, the ofregion between bond electron the strength densityconnection bonded established and pair in between [51]. thelence With internuclear the for power-law found oxides, wide use, the notfor only bond in a the valenceO’Keeffe study wide [52], of concept minerals, for class but hasrived example, also extended a of since the setvalence method solid of and and state de- bond bondfluorides, valence sulfides materials. length parameters and for Bresestrategy, nitrides. that a With they relate and a large clever bond nearly found number interpolation 1000 that of bondedbined oxides, the error interactions, of result parameters estimation,perimental with in estimated differences and a betweenevinced for estimated the small by ex- com- bond the largethe length number strategy of of together citations,acceptance with it by is its workers evidenttry in parameters that the for has study a foundrather of large wide than the crystal number radiifield chemis- are of maps used materials and incesses where the the as bond modeling, construction diffusion lengthsserved of for and that structure example, chemical bond of zoning.the correctness valence It such of provides was pro- ais also a structure. as ob- Indeed, useful reliability it asof check was using a argued sums of structure. that of radii it in checking the validity ( a large number ofpower oxides law can alternatively like be expressionwhere ranked of with the a form a given M– bond lengths, Fig. 1. by summingbonded the atoms) crystal plotted( radii with respect of to the Bonded interactions and crystal chemistry: a review This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 28 H et al. 22 -rings in 6 ) are vir- quartz-type  2 ) are virtually  126 ¼ ) [51] and those 144  ¼ begs a theoretical un- O disilyl oxide mole- –Ge 144 2 i O digermyl oxide mole- Si –Si 2 ¼ 6 –O– n/r cation. As such the bond –O bond length and bond h Ge quartz-type crystals can like- –O– 6 þ Ge– –Si and anti-tetramantane, C 2 4 ff Si– -rings as found in adamantine, and 6 20 ff –O– , , are virtually the same as those 6 H ; ) A s/r Si– 2 G. V. Gibbs, R. T. Downs, D. F. Cox 14 A ff –O bonded interaction is 1.0, exactly 1.77 ; 1.63 Si– A ¼ IV ¼ –O) and -rings with bond lengths and angles that clo- 1.61 Si would be 4, the same as the nominal 4+ 6 –O) IV –O) (M– ¼ R (Ge– (Si– , diamantine, C R O bonded interaction. In short, the bond number and R –O) 16 þ 4 H Just as diamond can be considered to be a giant mole- (Si– Si 10 R tually the same as those observed for a GeO number for an the same( as those observed for crystal quartz and nanometer-sized diamondmolecules consists molecules. ofconnected Each a C hydrogenated of framework these of inter- IV Connections between molecular and crystal chemistry The agreement betweenvalence the trends M– forconnection molecules and exists crystalsminerals indicates between and that representative thesicists molecules. a have Chemists bonded been and(cf. phy- aware interactions [59]). of For this of cules example, connection in and for a representative somephysicist comparison J. time of crystals, C. organic Slater thethe mole- [5] bond famous observed lengths nearly solidlike and 70 state years cyclohexane, angles ago observed (CH that for a tiny molecule the electrostatic bondbers strength are despite one andsuch, their the the fact same different that num- requirements, definitions each one for serves and an tomodel, ionic satisfy says origins. and the little the local As action. about other bonding Further, the for the a character well-developedbetween covalent of power the law bonded relationship inter- derpinning. observed for the gas phase H cule ( every direction, resultingmost in the rigid structure andcluded of his hardest diamond, study the materialreally a with known. molecule the Slaterthe of same observation [5] visible forces that acting dimensions, con- in ‘Diamond held a together is smallcule, molecule’. by crystal quartz and GeO the same as the electrostatic bond strength of 1.0 for an sely matchstructure those in observed thisstructure sequence of for of diamond diamond. where everthe the larger The cyclohexane type molecules of moleculeform. ultimate rigidity is The comprising is the resulting structure found is in braced by its rigid most C extreme observed for diamond. Asthan cyclohexane a is straight muchplacement chain more of rigid hydrocarbon, the he Hresult asserted atoms in that of the the cyclohexane generationlitany re- by of of C larger six-membered atoms molecules can C consisting of a C charge conferred on the Si wise beelectron considered density as distributionsare and giant virtually bond molecules the lengths samesmall and given as angles gas that those phase observedand their molecules. for angles representative Specifically, for the the bond gas length phase H cule ( among alternative positionsatom like that the valence of a metal i n h , con- i and n , defined h s as a func- n i –C bonds for –O) (M– and the longer the R –O bond length data h s , comparable with the 0.22 ) r / s 1.39( ¼ , the shorter the bond, the greater its represents the average number of elec- s , comparable with those obtained for i n 0.22 –O) h ) r (M– / i for any given bond strength, is comparable for R n h s/r, within the context of a closed-shell ionic model. It 1.39( s In a graph-theoretic study of the resonance bond num- Note that Pauling [23] was careful to use the phrase ¼ nected withanalysis the bonded ofR interactions the [57]. A data regression set resulted in the expression as the average number of bond electron pairs, the silicateabove molecules [53]. and As the oxide crystals discussed tron-pairs between thebe bonded taken as atoms,like a it the measure can bond of accordingly numbers the generated strength for of the the interactions, C– also supports Pauling’s [47]his perception discussion of of bond theedition valence electrostatic of in valence the rule. Naturethat In of ‘If the the the third Chemical bondsthe Bond resonate [23], valence among he the ofequally states alternative among positions, the the metalrule bonds equivalent to atom to the willpress the coordinated tend electrostatic atoms, the and valence toatoms’, satisfaction a rule a be would of discerning ex- divided wis statement the that graph conforms valences resonancenon with bond of the [50] analysis Le- and thevalence and the strategies. the nonmetal Brown Brese-O’Keeffe Brown-Shan- that and [52] Shannon bond bond (1993) valence length-bondlarger also correlates the found value with of covalent character, the hydrocarbons [23]. The agreement suggests that trend (Fig. 2b).expression Further, a regression analysis yielded the 6 expression obtained inand geometry-optimized the molecular analyses M– (Fig. 1b). of It thebeen is experimental found with noteworthy virtuallyvariety that the of similar same fluoride, exponent nitridetals expressions to and as have hold sulfide well for moleculesconcluded a and [55, that crys- the 56]. relative On change in the basis of this result, it was tion of oxide, fluoride, nitride[54]. and sulfide molecules and crystals bers calculated forsilicate representative structures, moieties it forbond was a lengths also variety of found correlate that with the the experimental bond number, ‘valence of theion metal atom’ in rather recognition than of charge the of the fact cat- that if the bonds resonate bond. are direct measuressity of between the pairsof of accumulation bonded of atoms, electron despite den- the definition ‘covalency’. They alsobond observed strength that trendscharacter the are of largely bond independent thetheory, length Burdett of bonded – and the interactions.that Hawthorne ionic bond [58] Using valence have perturbation theoryform since can of concluded be molecular picturedof orbital the as theory, interatomic a parameterized distance. veryry, by Within simple they the means found contextinvolving of that an the the theo- atom sumviding in of a an the stable analytical valencesOn structure basis the of is for basis the a the ofthe bonds constant, this bond pro- bond result, valence they sumwith lengths were rule. able in coordination tonumber, a establish number, that the stable the smaller structure larger the increase value the directly of coordination This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 7 b –O –Ge – br 2 –O . [67] re- (SiO), and the et al R , and the Ge– br –M bonded interac- –O –O bonded interactions organometallic complex is –O bond length, 10 (CO) 2 –O bond length, Ge– . [67]. The similarity of the properties of the ) of the bridging Si– a Mn, Co and Ni) for a number of organometallic –Ni bonded interactions for several Ni sulfides et al ¼ (Added in proof) In this elegant book “Solid and Surfaces” –Mn bond in the Mn 2 The close correspondence of the bonded interactions for ) of the bridging Ge– b ported that thedistributions physical and properties the of nature the of electron the density M– tions (M molecular complexes are virtuallylated the for same as anMn– those assortment calcu- of bulk metals. For example, the (VCH Publishers, Inc.(1988) NYC, asserts New that York,molecule pp. lying the 142), at chemist’sserved the Roald viewpoint heart solid Hoffmann of is statelocal, chemistry. often chemists But, but local the well theystrength, concepts with have have that packing the been been have considerations,energies, neither such several electrostatic ‘molecular’ concepts of forceswith as nor which ideas and ions, are that Madelung ionic unobservables. work, radii, and ‘What bond that can explain be structure wrong and properties?’ he isolated and shieldedsame by topological the properties ligandsdiscussed as yet below, the that the bond in bond criticalfor bulk point has the properties Mn the Ni– observed metal.are As also virtually themetal same and as those thoseGervasio calculated calculated for for the bulk Ni Ni metal complexes by cule has been used toof generate stable structures and for metastable aa polymorphs large random of number silica, arrangement starting ofbeen from Si found that and the Osity physical distribution atoms properties calculated [65]. for of It the thefor Si– has electron a also den- largecalculated number for of representative silicate silicates molecules [66]. are comparable withcrystals those and representativestricted molecules to is silicates not,dence and has however, also germanates. re- beenmetal A reported bonds to similar in exist molecules for correspon- classification a and of variety bulk metal-metal of metals. metal- bonds, In Gervasio a topological shielded metal-metal bonded interactionsthe metal-metal in interactions molecules in and tals the crystals again and suggests the bulkinteractions that me- in the the three forces systems behave that as govern if localized. the bonded a . et al ) [51, –O and  130 –Si angle data for the silica polymorphs [62] are plotted on (a) and experimental Ge– ), Laplacian ¼ r –O– ( quartz-type also r D 2 –Ge disilicic acid molecule –O– molecule calculated as a function ( 7 7 O Ge– –Si (Fig. 3a). The bond O 2 2 ff Si Ge –O– 6 6 , Si– A ff –O bond length and Si– –Ge angles for a variety of germanates are plotted on (b). The level line contour interval for (a) is 2 kJ/mol and that – 1.74 br ¼ –O bridging bond in the silica poly- –O molecule and likewise found that the bulk ) and electron localization function maps, 7 –O) c r –O) and O ( 2 r 2 (Ge– (Si– Ge R 6 R Potential energy surfaces for the disilyl oxide molecule calculated as a function ( –O bond in crystal quartz (465 kJ/mol) is virtually –Si bonded interactions for the silica polymorphs and –Si angle [54] and for the H ¼r The conformity of the experimental data with the poten- ) –O– –O– r ( of the bond lengthgermanate and angle crystals data observed including for a the variety of GeO for (b) is 6.3 kJ/mol. 60]. Molecular orbital methodstimize were the used structure to of geometry the op- H crystal ( Fig. 3. Bonded interactions and crystal chemistry: a review Si– tial energy surfaces andment the bond that energy supports thepolymorphs the argu- bonded are interactions virtuallya representative comprising the molecule [63].it the same Given may silica as this not correspondence, that those be surprising comprising to deformation learn, as electron demonstrated below, density clustered about thewithin point a somewhat of deeper minimum potential energy energy well but (Fig. located 3b). angle [62]. Experimental Si–bond lengths and Ge– in the generationtion of a of potential energy surfaces as a func- lengths and anglesare observed for plotted several(Fig. on silica 3a) polymorphs where the theest surface bulk lying of for the elliptically-shaped dataates level purposes scatter the line within of valley contour thecomparability floor that low- comparison of deline- of the thethe observed potential bond silica energy lengths polymorphsgenerated and surface. for and angles The the in theenergy disilicic acid potential of molecule energymorphs the indicates and surfaces that Si– the the moleculethe are Si– similar. Inthe fact, same the as energy of thatthis for observation, the molecule itlengths (462 kJ/mol) may and [61]. angles not Given are for be crystal for surprising quartz theduced are that molecule. the transformations the In same a bond for as later they the study germanates, of the Hattori pressure in- other silicates [64]. In addition, the force field for the mole- Si– generated for thesame disilicic as acid those molecule, observed are and virtually calculated the for the Si– L [62] calculated afor the corresponding H potential energy surface This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , , ) c c r r r ( and r et al. 1 )isa at r l can be ( , meas- j 3 c y r r l 2 . @ 3 i r l x @ þ )/ c 2 r . The greater the l ( c r r þ 2 is 0.0 (reflecting the @ 1 –C bond for ethylene, l ) perpendicular to the 3, ¼ c r j ¼ ( ; i ) r c H –CH -bond, typically the case for r 0.45 (reflecting the multiple . The strength of the interac- ( s c C– r ), ) is negative, the value of 3 r 2 c r ¼ perpendicular to the bond path, ) parallel to the bond path and ( r c ( r r -system, the cross-section is ellip- e c r r 2 , ( –O bonds in a silicate, the cross r -orbitals (one on each C) that can -character. But, with the superposi- s G. V. Gibbs, R. T. Downs, D. F. Cox r p r s toward the bonded atoms, the greater i.e. c , r c ). In addition to providing information r c , Bader’s strategy provides a straight for- ), at the bond critical point. By evaluating c r fashion, c ( bond) whereas the C– r where r r ( )at r p p 1), of the cross-section of the bond path. In r r ( r , with two also provide information about the shape of the 2 2 -bonds on the 2 l -bond for ethane, H l p / s 1 l –CH ( As demonstrated by Bader and Essen [73], During the latter part of the last century, Richard Bader By focusing on such topological features of the elec- and C– –C , measure the curvatures of 2 ¼ 1 2 . At a point H difference in theire magnitudes, the greater the ellipticity, the extent to whichtrated the parallel electron to densitybonded is pair the locally of bond concen- electron atoms, density the path at greater in the concentration the of direction the of the bond critical point denoted l ures the curvature of ward method forproperties studying and of characterizing the physical bond path andlocally the concentrated extent at to which the electron density is absence of a the shieldingelectron of the density.form Further, nuclei of the by theeigenvalues Hessian the trace which equals locally in of the turnplacian algebraic concentrated the of is sum equal of diagonalized to the the three value of the La- the greater theirtration magnitudes, of the the greater electronpath. the In density local contrast, toward the concen- the positive axis eigenvalue, denoted of the bond they found that theconcert with three the real valuekinetic eigenvalues of of and the potential the electron energy matrix density density in and properties the at local l cross-sectional distribution ofured perpendicular the to electron the density bond path meas- at overlap in a character and the asymmetrybution of about the the electronbond density axis is distri- of unstablecross-section the of and, bond the say, bond [2]). is on Further, typically the highly when elliptical. verge a of rupture, the tion between aregular given way pair with of thedensity bonded progressive in atoms accumulation the increases ofpoint interatomic in as electron surface a the and equilibrium bond at length the decreases bond[2] [2]. critical and hisstrategy colleagues for forged studying a minimumbased powerful energy and on bonded widely the interactions distribution, used physical properties of the electron density tron density at about the local concentration of the electron density at the Hessian matrix of used to characterize aeigenvalues bonded defined interaction. Two bythird of the the is three Hessian positive. The are negative negative eigenvalues, denoted and the the case of athe well-developed majority ofsection is the typically M– circular,a indicating large that component of thetion bonds of have useful probe foris determining locally whether concentrated the orr electron locally density depleted at a given point tical in certainC– cases. For example, the ellipticity of the ), r ( r is the unit n ) suffices for the r ( yields the total num- ) is defined by the r r [70]. r ( ) where r S S ( )d S r ( r 2 Ð r 8 0 ¼ ) such that r ( r n Á ) can be viewed as the number of electrons ) r ( r giving the amount of electron density in the ( r r r ) is also a single-valued scalar field defined over r r )d ( r r ( r A pair of atoms, connected by a minimum energy bond equation ber of electronstron for the distribution arraysaddle constitutes [68]. features The local form thatforces maxima, of operating are the minima within adefining elec- the and physical the array manifestationhighly [2]. positions accumulated of The electron of local the in density maxima, response nearly in of space, sphericallocalized the forces are distribution domains exerted to formed bydomains the the of are highly nuclei. not attractive Sufficethe it and only to nuclei, centered say, but at the ture they or of also near the determine thetions bonded the positions of array. the of bulk In arrayindividual of in contrast, the electron concert the struc- with bonded shieldedformation the of interac- forces bonded minimum exerted energy atomssity by bond that the result paths connect of the insity electron bonded along the den- atoms these [69].of The paths any electron is neighboring den- a line.a maximum It surface also with results respecttween defining in the to an the bonded that formationdle atoms interatomic of point that zero-flux where intersectsmum the surface the value. electron path be- density at A a adopts zero-flux sad- a surface local mini- with volume element d 3-space [2]. determination of all of[3]. the In physical addition propertiesspace, of to the being array positive definite everywhere in for a bonded arrayin of the atoms net adopts forcesthe a on configuration each energy where- ofWhen of the atomic these the nuclei conditions are resulting are zero configuration and fulfilled, is minimized. asks. What isconcepts wrong, may draw orheart that can field, of be that chemistry, groupwhich wrong, the of permits is scientists, a molecule. thatdiscrete away connection ‘One from application molecule, between must the of organic thechemists select such structure or have the at inorganic’. isolated handeven explanation themselves As inorganic and (no such colleagues some wonder aren’t ‘manying interested that not solid in their to what state see organic they bonds do) and in by their choos- materials’. vector perpendicular to the surface path and sharing aered zero-flux to interatomic be surface,contour bonded is consid- map [2]. When isclei a constructed of level in a line adefined, pair electron plane often of density containing exceedingly bondedor the high atoms, nu- local it very typically maximalinked near displays centered by the well- at alowing positions very the low of bond lyingand path. the ridge define Again, nuclei of the themanifestation electron positions of local of density of maxima the the fol- tron represent the bond atoms density atoms path that and oferal, connects the minimum the the ridge bonded energy valuepath is atoms elec- decays of [71]. a monotonically In theapproximately between gen- electron the piecewise bonded densityadopts pair along exponential a in local each an minimum fashion bond value at [72] a point and referred to as the In the ground state, the electron density distribution, Electron localization function, deformation and Laplacian of the electron distribution 8 This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) 9 r ( L ) and the r ) model to ( r L ( ) distributions L r ( h ) electron density r ) maps can be ex- ( r ( r h D ) and r ( L ) for a homogeneous gas which r serves as a reference system and ( . [82] found that a close connec- C r) ( h ) model and domains of locally con- et al r C ( h , the fully delocalized reference system. 5/3 ) ) distribution maps. A deformation electron r )) such that the electron localization function r ( ( r ( r r h D In a careful consideration of For the rock forming minerals, maps of ) maps are in close correspondence with the domains r ( in force [80], and Inasmuch as thepact Pauli on principle has thespins, a kinetic the relatively energy value small forcase of im- electrons can the be with excess expectednator anti-parallel local to of kinetic be energy small in in value this (in the denomi- can be expecteding to bonding be and relativelyelectrons nonbonding large with electron-pair for anti-parallel domains regionsthe spins where kinetic harbor- are energy localized. is Inelectrons expected contrast, with to parallel be spinstrons large reside, in with particularly regions since where about parallel elec- one spins order ofparallel tend magnitude spins greater to [80]. thanspins, repel those Accordingly, with the for one anti- kinetic regionslarge another with energy and by parallel the densitysmall. electron can As localization be asserted cantrons expected by be for to expected Gillespie bondeddomains be to and system of be Hargittai tend anti-parallelactions [81], spins to of as elec- be a determined localized bonded by in system. the disjoint Thus, inter- represents the value of varies as maps display similartron features, density with local accumulatedthe maxima nonbonded along of regions, the elec- featureslone bond also electron ascribed vectors pairs to [8, and bond 83–87]. in and To appreciate the similari- electron localization function depicttures easily of understood domains pic- trons ascribed as to discussed bondingsince above. and the In lone last addition, pair century, as elec- deformation used extensively pected to showwith local anti-parallel maxima spinsand in in are regions lone localized where pair regions. along electrons bond vectors for a number ofinteractions, molecules Bader covering ation spectrum exists of bonded betweenprovided the by the domains of electron localization maps [82]. Also,the number a and close locationthe of correspondence electron the localization exists local functiondensity maxima between and displayed deformation by electron density distribution, definedtotal as the density difference forof between a the the crystal density,used and extensively is a in a procrystalactions the representation well past in in knownalong the terms function bond study vectors that of of anda has bonded in procrystal accumulations inter- been nonbonded representation of regionsspherical corresponds [83], ground-state to electron where atoms thethe density located corresponding sum atoms at of occupy the the in the positions crystal. that centrated electron densitythe provided extent that by therelated the two in distributions are afields homeomorphically are topological similar sense.of in terms the In of domains otherthey the of number define. words, localized and In and thedomains arrangement addition, concentrated two the of density number that h localized and electron positionsof of density locally the displayed concentrated by electron density the displayed by . , . r r c 0, r < ! 0, the ) s ) r ( r < ( r ) 2 r c 0, the elec- = r r 2 ( j . As stated by s c > r are their occu- 2 r , i i ) ) c n r r ) contour map for r ( r . In this case, the ( ( is negative and the r r r L 2 r and jr ) for the electron den- r 8 r 0, regions are mapped = ) is necessarily positive, 3 ( 1 r l r ( 2 Þ – r > 2 2 > 2 j ފ ) j r r s ) contour maps displaying the r 2 , ð i r ( centered at l ( h r r) L 2 r ( C Þ¼r 2 þ = r j w r Þ ð s 1 ), the electron localization function, , r i L r l ð ) ) describes the difference between the jr ( j i r r C L ( ( n C w s j þ½ i ; i P n 1 ð

are the orbital densities and i 2 = P = s 1 1 –Si bonded interaction for the disilicic acid mo- ) r ¼ ( ¼ s r ) –O– Þ¼ ) r r r ð ( The function C( h ), has also played an important role in the study of the In addition to r In a mapping of r ( When the conversetron is density true is and positive the and curvature of the elec- electron density is said to be locally concentrated at tron density is said to be locally depleted at kinetic energy density ofenergy the of actual the system system and without the the kinetic Pauli exclusion principle pation numbers. electron density is said to be locally concentrated at where h domains of localizedThe function, electron defined density bybe expressed Becke for as and minerals Edgecombe [78]. [79], can lecule), the numberswere and found the tothe relative correspond bonded positions with andmodel of nonbonded the for which electron closed-shell domainsered molecules pairs ascribed for [76]. of to It a thelocal was variety VSEPR maxima also of concentration discov- molecules andsity for depletion can which of the be electronand domains den- connected nucleophilic of withthe attack, known maxima respectively, sites correspond demonstratingattack of to and that electrophilic sites thecorrespond of with local potential sites minima of electrophilic potential (holes nucleophilic in attack the [77]. valence shell) the Si– shell structures for isolatedcal Si combination, and O however,locally atoms). concentrated the Upon electron chemi- uniformcally density valence in distorted the shellFig. with figure 7 of in is the [75] typi- for creation relief of and level local line maxima (see sity distribution for anwas isolated discovered first [74] orstructure that second row the of atom, distribution it sponding the mimics number the atom of shell pairslocally faithfully of concentrated spherical by and shellscentered locally of displaying about alternating depleted the a electronfor nucleus corre- of density relief the and atom level (see line Fig. 5 in [75] where is necessarily greaterof than a it is sphere on of average radius on d the surface Bonded interactions and crystal chemistry: a review curvature of the electron density at but when the converse is true and and the electron densityAccordingly, when is said to be locally depleted at Bader [2] ‘By mapping those regions where one is mappingdensity those makes regions its whereof dominant the the contributions potential total tothose energy energy the regions of lowering where the system’. Further, by mapping where the kinetic energytion density to dominates the in energy its of contribu- the system (see below). This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) r ( –O –Si [75, r et al. D 4 )map r –O– O ( 4 h –O bond angles for Si 8  –Si angle. The and H –O– 7 triangle of the three (r) isosurfaces [86]. plane adjacent to the O 3 r 2 –Si plane, capping the 3 D Si 6 0.83 colored gray and –O– –Si angles comprising the ¼ –Si line [86]. Figure 4b dis- –O bond vectors but with a ) O, H r 2 ( –O– –O– ) (Fig. 4a) represent cross sec- ) [92]. It is noteworthy that the h Si r ) maps for the 180 3 6 ( r ( r A G. V. Gibbs, R. T. Downs, D. F. Cox L D –Si line. Overall, the agreement be- ) than those recorded by the experi- ) color contour map generated in the 3 r ( A ) maxima are located along each of the h –O– r 0.40 e/ ( ) and –Si angle also shows maxima along the r r ( D h –O– 0.30 e/ –O bond vectors that define the plane [94] ) isosurface maps generated for the five nonequi- composition. The maps for the four nonequivalent r ( 0.85 colored pink) exhibit similar features with 2 –O bond vectors with local kidney-shaped features –Si angle [88]. The distributions are also similar to h ¼ ) isosurface envelopes that enclose the local maxima Unlike coesite with four coordinated Si, the Si atoms ) –O– –O bond vectors, but unlike the bent angles, two low r r ( ( three Si– (Fig. 5a). Local maximaregion are on also opposite displayed sides in of the lone the pair OSi Si atoms coordinating theretical O local atom. Experimental and theo- maxima along themore bond electronegative vectors oxide aremaxima anions displaced recorded and toward thesmaller for the heights ( the of theoretical the maps are typically 91, 92] and for quartz [93]. In general, the local angles. They aresilicate likewise molecules similar like H to those calculated for bent angles areh strikingly similar(referred to to as isosurface thethe maxima) Si– well-developed displayed alongcapping each the of oxideSi– anion onthose the reflex reported sidesingle for of chains the of eachwollastonite-2M Si– silicate bent tetrahedra [90] in andseveral spodumene the fibrous [89] zeolites and tetrahedralsity [91], frameworks extending with for into ridges the of interior electron den- of each of the Si– oxygen atom. A heights offound experimental to andlengths. be The theoretical independent lonecontour pair maxima of maxima mapping displayed the were of by observed the level Si– line mental maps ( both coesite anddeveloped the ring disilicic torus isosurfaces aciddicular about molecule to the display the Otween well- atom Si– the perpen- two distributions is comparable. of the highbonded pressure to silica polymorphthree six Si stishovite oxygen atoms, are forming atoms each a where planar OSi each is bonded to link the corner sharingof silicate SiO tetrahedra into a framework tions through kidney-shapedoriented perpendicular isosurface to domains the that Si– are oxide anions and bisectingstraight the reflex Si– Si– Theoretical generated fordual the isosurfaces straight alongbroken the angle set Si– of displaysrather isosurfaces than hemispherical forming wrapped a aboutthe torus-shaped the domain experimental as oxide and displayed anion by theoretical Si– lying lone pairoxide features, anion are located displayed, ona defining circular opposite a torus cross sidescular section shaped of and through lone the surrounds pair theplays domain Si– that isvalent perpendi- angles. Thefour dual isosurface bent mapsh generated angles for the well-developed (for hemispherical domainsbond along vector each andoxide Si– kidney-shaped anion on isosurfaces the capping reflex each side of the angle. The b with 3 ) and carbo- r ( A ) contour 3 L r –Si angles ( r –O– D –Si angles that –O– ) distributions will be r ( h ) (referred to here simply as ) contour maps presented in r r ( ( polymorphs, andalusite and sil- h r ) and 5 a D r ( ) electron localization function, ELF, iso- b r SiO 2 D ) Level line deformation electron density a ) maps for predicting chemically reactive sites such ( r ( h An aspherical-atom multipole modeling of the theoreti- cal electron densitythe theoretical distribution level for line coesite [86] yielded nate magnesite [88]. In addition, the use of the isosurface maps) generated forrock the forming valence electrons mineralssilicate of talc the tremolite, will diopside alsobond be and and examined lone the in pairplayed terms localized sheet by of electron the density thenonbridging ascribed domains oxygen bridging dis- atoms oxygen of forsterite atoms [78]. of quartz and the the as centers ofsites of potential protonation electrophilicbearing for attack Si stishovite deficient andiso-valued will coesite preferred surface and maps be H of examined. and Al Three-dimensional Fig. 4. maps [60] compared withsurface ( maps calculatedof for coesite. the The five level nonequivalent line Si– contour interval for (a) is 0.05 e examined below for thestishovite, two the silica two polymorphs Al coesite and ties of these maps, 10 the positivedotted contours and drawn therepresents as zero an one solid ELF is0.85. lines, value dashed. The of The the red 0.83 graySi. spheres negative and colored represent ones the ELF O pink isosurface atoms are and one the an blue ELF ones value represent of limanite, the borosilicate datolite and the MgCO Fig. 4a for the five nonequivalent Si– This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. c b d 11 . [96] gener- octahedra paral- et al 6 with the chains cross-linked c polymorphs sillimanite and andalusite 5 SiO 2 a b a The two Al lel to thetogether unit by cell fourby vector coordinated four Si coordinated andsite. Si Al In and in a five study sillimanite coordinatedof of the and Al the six in coordinated electric andalu- Al field atoms, tensors Dahaoui at the positions each contain chains of edge sharing AlO tions in stishovite consist ofshell a interactions. mixture of Further, shared themajor and calculations closed- changes indicate occur thatcreasing pressure no in between zero the and 150 bonded GPa. interactions with in- ated experimental static deformationminosilicates, maps using for an theelectron aspherical-atom two multipole density alu- modeled distributionsingle obtained crystal with diffraction high data. resolution Figures 6a and b show level ) r ( r D –O bond vec- ) cut in , the po- ) level b 3 r ( r plane, capping the oxide anion as A ) and andalu- D c 3 ) maps. It is apparent that the r ( ) together with from the plane are displayed together in r a D A ) paralleling (110) of b 1.5 ) and andalusite ( a ) level line contour maps for A model experimental defor- ) distributions exhibit the same number of bond and r ( r Model deformation electron den- ). The contour interval for the ( r ) maps that are comparable with experimental maps d h ) contour maps is 0.1 e D r r ( ( r r revealed by the Fig. 6. sity sillimanite ( a plane parallelingELF (110) maps compared for with sillimanitesite ( ( D plane of the trianglethe and a region dual isosurface map generated in mation electron density Fig. 5. combined ELFcontour isosurface maps ( andstishovite. Each color Si atombonded in to the four planeis O is bonded atoms while toure each three 4 O Si legend atoms.atom for See the color level Fig- code line and for information. the thedisplayed The in ELF (b) colorbar color is at contour the given map bottom by of the figure. color Bonded interactions and crystal chemistry: a review line contour map ( sitive contoursones are are solid, dashednot and the the drawn. zero negative for The contour the is color ELFbar contour map graph is interval Fig. given at 4 by legend the the forformation. color the bottom The ELF green ofAl isosurface and in- spheres the (b). red represent ones See represent O atoms. determined earlier with acrystal combination of data powder [84]. and The single in band the structure maps were and taken lone as pair evidence features that the bonded interac- and Fig. 5b. Hemisphericalmaxima dual are located isosurface along and eachtors. of color the In three contour addition, Si– both dual sides isosurface of the local OSi maxima occur on lone pair domains. In antions earlier and study of the themented elasticity bonded interac- of plane stishovite,D using wave linearized calculations, aug- Cohen [95] generated This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) r ( –O –O r car- –O5 et al. –O4 D 3 ). c –O4 and , is a car- ) O4-B– 3 –Si– b ) color con- r –O bond vec- ( h ), respectively. See –O5 bonded interac- d ) the O3– a ) level line map de- r ( –O4 and the ( Level line static model de- ) O4-B– r ) and ( b c D –Si– –O bond vector and each O maps for ( the ( tions for datolite.vicinity Contours of inomitted. the atomic Color positionsELF contour are andO3– dual isosurface maps for bonded interactionsin are displayed ( Fig. 6 legend forthe information about lineSee contours Fig. in 4 legendsurface (a) for and the ELF information.spheres (b). iso- represent Theones represent Si, B blue represent and the O the atoms. red green contour The ones ELF interval color iscolor bar defined at by the bottom the of ( Fig. 7. formation electron density G. V. Gibbs, R. T. Downs, D. F. Cox d b ) contour maps, with hemispherical Ca r ( –O bond vector that radiates out of the Ca r D –O3 bond vectors and along the B– O2 H –B bonded interactions (Fig. 7d) as observed for –O vectors are in the direction of the lone pair feature. Unlike the other minerals, magnesite, MgCO –O4 and Si– –O– bonate with acubic structure close that packed canin array be one-third of viewed of Oatoms as atoms the coordinated a with available by distorted the three octahedral Mg O voids atoms atoms, and forming the a C CO conform with the experimentalface maps, forming with a theSi– bracelet dual isosur- about O4bond with vector maxima that alonglevel radiates the color out mapsplayed of likewise the by conform plane withdual the isosurfaces (Fig. the 7c). shown maxima along The tor dis- three and of along the aplane. B– Kidney-shaped Si– isosurfaces existSi– perpendicular to the danburite. Also, asCa– observed for danburite,As such, several the of kidneyfor the shaped domains the appear Ca tohave atoms be also in attractors been datolite observedLikewise, and for as danburite. the Similar wecoesite K results shall and atoms stishovite see in appear later, wadeite to the be [78]. attractors lone for pair H. domains in bonate group. As such,and each a O atom Cand is atom. bonded Pavese to [99] Usingpicted two generated Mg theoretical in the structuremaxima Fig. 8a factors, occur for along Catti atom each the is C– coordinated carbonate bybonded two group. region maxima that of Well-defined tour reside the and in atom. dual the non- isosurfacedisplays A maps combined similar for thebond features carbonate vectors molecules with andever, in maxima the the along maps lone differ the pair in region C– that (Fig. the 8b). maxima How- along the C– 6 –O4, –O4 plane of –O3, Si– c a –Si– ) map for sillima- –O5 plane for the –O2 bonded inter- r ( r Al– –B– D IV Al atom. The distribution corner sharing tetrahedra IV –Al1 angle, as observed for 4 B ) electron density extending to- –O– r ( r D OH), is a sheet structure consisting 4 and SiO 4 –O5 bond vectors. The level color maps ) maps show three maxima about O2 including ) dual isosurface and color contour maps shown Al atom. Local maxima occur along each of the r r ) maxima are located along the Si– V –Si angle. Also, a local maximum occurs along ( ( r group (Fig. 7a) and the O4– ) contour maps for the two minerals in planes con- h ( h r –O bonded interaction. The map for sillimanite is 4 ( r –O– r D group (Fig. 7b) where O3 is bonded to a B, a Si and Al– ) level line contour maps for the O3– D V r 4 ( Datolite, Ca(SiBO B –O4 and B– r Ca B– linked togetherhouse in Ca four atoms betweenO and the atoms. eight sheets Ivanov inD and membered cubic Belokoneva antiprisms rings [98] of the recently that SiO calculated BO similar to thatbonded to observed two for Al1 atoms andalusite and an except that O2 is taining the Al1octahedra. atoms In and the the casebonded of shared to andalusite, edges two the Al1 of oxygenAl1 and atom the and a O2 a Al1O Si is atom and O1 is bonded to two line 12 0.0 ELF 0.98 action. With this exception,mappings the is agreement comparable. between the two bond vectors, with the one along the bond vector along the of the maxima aboutthe O O2 atoms is for similarma. andalusite The to but that O2 displayed onlyfor about displays O1 two and maxi- O2of for sillimanite andalusite (Fig. 6d) (Fig.of 6c) are the similar, and with bond those maxima vectors aboutnite, along [97]. O1 the each Unlike the of alternating BO two Ca atoms,and O4 O5 is is bonded bondedoped to to a B, Si, H a and B two and Ca a atoms. Ca Well-devel- atom the Si– the ward the interior of the Al1– This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. b 13 ) maps r ( –Si angle , are pre- r ) dual iso- r 18 D ( O –O– h 6 Si [104] with well- 3 maps for diopside 6 Þ Be O 2 r –O bond vector. Each O 2 ð –O bridging and non- L –O bond vectors. a ) and the experimental , and beryl, Al r ( 18 h O 5 Si 4 –O bond vector and mushroom shaped iso- ) displaying the six next nearest Mg atoms (violet b Al 2 ) displaying four next nearest Si atoms (blue spheres) and a ) isosurface maps generated for the quartz and for- r –O bond vector whereas each nonbridging O atom of forster- –Si reflex angle. Features similar to those displayed ELF dual isosurface maps for the silicate tetrahedral oxyanion ( –Si angle and a hemispherical isosurfaces is displayed along h Given that the The dual isosurfaces for the framework silicates cor- Prior to the calculation of the dual isosurfaces maps for –O vectors for both minerals. In the case of quartz, a –O– –O– for the foregoing minerals are comparable, defined local maximabridging along bond the vectors Si– andO3 atom. in the lone pair of thedierite, bridging Mg Fig. 9. in quartz ( for forsterite ( spheres). Each bridging Ois atoms capped of the bySi– tetrahedral a oxyanion of kidney quartz shaped isosurface on the reflex side of the surface maps wereforming generated minerals. for Thea maps a mixture in variety a of ofatoms number the of other of isosurfaces quartz rock cases and displayedThe exhibit the by nonbridging the Osterite bridging atoms structures of O are forsterite. spherical displayed in isosurface Figs. domainSi– 9a and is b. foundkidney-shaped A isosurface hemi- along domain eachof is of located the in O the the atom vicinity on the reflex side of the Si– each Si– ite is cappedspherical by isosurface a displayedis along mushroom each shaped represented Si– isosurface byabout again the a isosurfaces. with red a sphere). hemi- See Fig. 4 legend for information diopside, tremolite, andtry-optimized talc, with their the structuresbond VASP were [100–102] lengths geome- and software anglesperimental and were values the within found to fewtures percent agree each with [78]. contain the As moietiesforsterite ex- and these similar quartz, struc- in the structurethree isosurface display to maps features those generated similar in ite for to and the those quartz exhibitedserved by (Figs. for forster- 10a, forsterite,along b hemispherical each and Si– isosurfaces c,surfaces are respectively). with found As threeatoms. ob- local Also, as maxima observedis cap for capped the quartz, each by nonbridgingSi– bridging O a O kidney-shaped atom in isosurface Fig. that 10b have bisects[103] been the and reported the for clinopyroxene LiGaSi whereas in thebonded to case a of Siby and forsterite, three a where Mg mushroom atoms, eachdirected each shaped along O O each isosurface atom of atom with the is three capped is the Mg– local maxima ). b a b –O and B– III ) map for the r ( r D –O, , the solid, dashed and 3 Al– VI A –O, Si– VI ) maps are similar in magnitude ) maps are substantially smaller r ) map is 0.1 e r ( r ( ( –O, r h r D D Si– IV Static model deformation electron density ) and a combined ELF isosurface and color contour map ( a –O bonded interactions each have a component of –O vectors in the C– Contour interval for the carbonate anion in magnesitetors generated ( with theoretical structure fac- shared covalent character [83, 88]. than those inthe the nonbonded nonbonded maxima region. areMg It oriented atoms is in like the the noteworthyite attractor direction that [78]. nonbonded of The domains the tors local in danbur- and maxima in displayedevidence the along lone that the pair bond the regions vec- for the minerals serve as III to those in theC– nonbonded region whereas those along the bond vectors in the Fig. 8. Bonded interactions and crystal chemistry: a review dot-dash level lines representrespectively. positive, See negative and Fig.faces. 4 zero The valued legend lines, colorred for contour spheres information interval represent O on is and given the the at green ELF ones the represent isosur- bottom C. of (b). The This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) c for for –O et al. c a –O bond ) and tremolite ( b octahedra in beryl 6 –Si angle in cordierite –Si angle as observed in ), talc ( a ELF dual isosurface maps for –O– –O– –O bond vector as observed for Si– –Al angle in sillimanite [97]. –O bond vector. An examina-  viewed down the unit cell vector diopside and tremolitetalc. and See down Fig. 4mation. for The the isosurfacethe red infor- green spheresand ones represent the O, Ca, violetnonbridging the ones O blue Mg. atomsmushroom Note ones are that Si shaped capped the served isosurfaces with in Fig. as 9beach for bridging ob- forsterite Oshaped and is capped that isosurface by oneach a the kidney reflex Si– sideFig. 9a of forisosurface quartz. isthe displayed An Si– along hemispherical quartz each and of forsteriteclosed and each by Hface. a is roughly en- spherical isosur- Fig. 10. diopside ( –O– 180 G. V. Gibbs, R. T. Downs, D. F. Cox O O O O b c Si Si Si Si H H H H octahedra in cordierite are bonded to Si, Al and –Al bonded interactions comprising the six-mem- –Si angles in beryl are each capped by a kidney 6 Mg Mg Mg Mg –O bonded interaction. A similar mushroom shaped –O– –O– Mg. They likewiseas exhibit in mushroom beryl,rical shaped with domain isosurfaces two along maximation each in Si– of addition the tothat a isosurfaces as hemisphe- for thefrom two coordination a to number three variety totends of four, to of the that progressively the change O minerals from isosurfacekidney-shaped atom two about isosurface shows hemispherical the increases as and atom in a rical quartz and to a a mushroom single shapedima isosurface hemisphe- with two and local max- finallyshaped isosurface to with one[78]. three local hemispherical maxima and as in a forsterite mushroom is surrounded in partabout by the lone O1 pair atomSi– features in similar coesite. to Thebered those isosurface ring maps in forgion the cordierite but lack show a featuresbonded hemispherical interaction in isosurface and the alongAl– a lone mushroom the pair shaped Si– re- oneisosurface along was the recentlyvector comprising reported the alongThe Si– the O Al– atoms comprising the AlO sented in Figs.Si– 11a and b.shaped The isosurface O asatom atoms observed comprising comprising for the the quartz whereas the O bonded to Si,isosurfaces, Be but and with Al,as two also observed rather display forMgO than mushroom shaped forsterite. three local The maxima O atoms comprising the ) b a b –Si an- –O– Si–  ) and cordierite ( a a . The Be atoms in beryl are co- c ELF dual isosurface maps for beryl ( viewed down the unit cell vector Fig. 11. lored violet, Al atomsMg green, atoms Si in atomsinformation. blue, Note cordierite O that is atoms the are violet. isosurface red See about and the Fig. the 179.6 4 legend for isosurface 14 gle in theplayed six-membered about ring the of straight cordierite angle is in very coesite (Fig. similar 4b). to that dis- This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . 4 b 15 repre- et al silicate v 4 ) isosur- r ) and the- ( . For pur- r h ( A defect ( r v D Four H atoms are v. oxyanion through the 4 oxyanion is displayed for a 4 ) local maxima, H was pre- ) and an ELF isosurface map for the r through the local maximum of the ( a –H vector is oriented perpendi- ) maps for coesite, Gibbs h a r A –Si angles. The yellow spheres in (a) ( from the O atom for Al-bearing plane [84, 86, 94, 95]. Given the h 3 –O– A ) and r ( L ) isosurface maps for stishovite also display r defect. As the Si2 atom is bonded to O2, O3, ). The silicate polyhedral in (a) are colored blue in (a). ( b h 4 –Si angle. A drawing of the proposed geometry of an As observed above, experimental model –O5– oretical local maxima inboth the nonbonded sides regions of of the the OSi O atom on gant analysis ofaverage, the replace FTIR onean of spectra the (OH) that Si2 four atoms H in atoms, the on formation of O4 and O5,the the O H atoms atomsbasis involved are of in accordingly the only theestablish bent bonded spectra, that to angles. the thedetermine Further, authors four on the the were H positionsdefect, not atoms of a only replace the remarkablestudy Si2 able H accomplishment. of atoms but to the In of also an the to independent resulting [64] predicted the positions of the H atoms of the (OH) defect by extending theof vectors the from four the positions O of atoms each of the Si2O Fig. 12. sents the vacancy) formed by a missing Si2 atom of an Si2O The white spherepredicted in to (a) be representsvectors the disposed directed vacancy, about atthe the the vacancy reflex maxima with siderepresent of their of the the potential the kidneyplayed OH local shaped Si– in maxima isosurfaces (b)of on of and the the the H green atomsvector kidney spheres (a). from The represent shaped position the eachdefect of isosurfaces predicted of each at positions dis- H the a was positions found distance by of of extending the a 0.98 O atoms that houses the tetrahedral oxyanion in coesiteoxyanion ( ( kidney shaped isosurface in forming an OH bond. maxima of theposes domains of at illustration, a the distance Si2O of 0.98 positions of onedicted of to the dockstishovite such 0.96 that the O– representative moiety oftogether the with coesite theand positions structure the of predicted in positions the Fig. of lone 12a the pair H local atoms. An maxima face map ofin the oxyanion Fig. 12b istween displayed where next the an toatoms, nonbonded the one-to-one their moiety kidney-shaped mapping localthe domains maxima is H of and mademined the the positions. be- H predicted O The positions positionsthe agreement and of lone between thosethat pair determined the no on maxima IR effort therelaxation is deter- was basis of comparable, made of thethe particularly to structure defect. correct given Notwithstanding that the thetion, would failure positions surely it to for make occurfound may any the about to correc- not be theSi– be best for surprising the that H atom the involving the agreement smallest was ,  4 SiO 2 -Mg b ) maxima of the non- r ( h ), is indicated to be the most  –Si angle exhibiting the largest –Si angles ranging between 137 angle, it is expected to be the least –O5– –O–  ) maps show that the magnitudes of the . [111] concluded on the basis of an ele- r ( h et al –Si angle [64]. As the angle decreases from ,  ), likewise increases progressively as the angle r –O– ( , they are asserted to be more reactive sites and ¨ller  L to 137  In a study of the incorporation of H in Si deficient The magnitudes of the local 180 and 150 Using the constraint thatthe the bonded sum interactions ofstructure reaching the each satisfies bond O strengthsatom, the atom of Donnay in ionic and aidentifying valence stable Allman those requirement [105]likely O of devised to atoms the a bepredicted in strategy protonated. that a for With the structure the O(1) that strategy, atom are Smyth in most [106] wadsleyite, Potential sites of electrophilicand attack protonization Bonded interactions and crystal chemistry: a review coesite, synthesized atKoch-Mu high pressures and temperatures, susceptible O atom in the structure for protonation [87]. bonded domains displayed byof coesite the depend Si– on the value was the most likelythat candidate was for protonation, corroboratedpotential a carefully by prediction generated agle with mapping crystal experimental of diffraction X-rayof data sin- the the [107]. electrostatic With bondsince his strength asserted successful sum use easy that to constraints, “the identify from Smythstrength protonated X-ray calculation”. [108] data oxygen by However, has where is a for the simple structures relatively ionic Pauling valence like bond valent requirements O coesite for atoms the areis five all the nonequi- satisfied, moston he likely concluded the atom that basis inwith the of a O1 the the purely structureelectrostatic electrostatic ionic to site model site be potentials [109]. potentials protonated quirements Although and calculated may the the have use bond beenatoms of strength successful in the a sum in structure identifyingtonation, re- that the they are O do potentiallythe not susceptible coordinates provide to of information pro- theOH for vectors H determining or atoms both. and the orientations of the reactive O atom inceptible the to structure andnonequivalent protonization. as O such In the atomsinvolved contrast, least in (O2, sus- as bent O3, Si– the O4 remaining and O5) are each maxima progressively increaselocal in concentration value.comprising In of the addition, lone theplacian pair the electron domain, as densitydecreases displayed with by distribution the the Si– La- local maxima [110]. Withtration the of increase the in ED,to the the increase local reactivity with concen- of decreasinggressively the angle, more O rendering reactive atom thetrophilic and is atom indicated susceptible attack pro- to [74,involved potential in 77]. elec- the As 180 the O1 atom in coesite is hence more susceptiblecause to the protonic smaller attack theof than angle, the is the O O1, greaterto atom be- the protonic and electrophilicity attack. Moreover, thetal a greater mapping deformation the ofthat electrostatic the apparent the experimen- susceptibility potential deepest electronegativenext for to minima O5. coesite in As theangle shows such, maps in the occur the O5 atom structure involved (137 in the smallest This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. –Si et al. predict –O– a priori formula units, the atoms of 2 . [118, 119], starting with a ran- G. V. Gibbs, R. T. Downs, D. F. Cox derivation of the structures that can et al –O bond lengths and the Si– 1 space group symmetry. Collectively, the P a priori With the development of local density functional In an effort to achieve Hawthorne’s first goal, a deriva- can crystallize with awith given chemical a composition,Hawthorne starting [28] random observed that arrangement ‘We cannot of atoms. At the time, crystallize in atry given chemical optimization system ofresulting and a (2) in structure, the starting athat geome- with rivals structure those a determined and model experimentally. properties and with(LDA) an theory, substantial accuracy geometry progress optimization has and the beenof simulation made materials of in [3]. theour Not properties the understanding only of have theminerals the properties in calculations and the advanced transformationsalso Earth’s of advanced interior (cf. ourtween [115]), the understanding but propertiesand of they of structures have the the inenergy connection electron terms density be- density distributions. of Inat distributions the studies the local of high kinetic mineralsstructures, equations formed pressures and of potential state, encountered elastichave in and transport been the properties computer Earth’sphases as modeled interior, olivine, for wadsleyitewe and such will perovskite. restrict In crust our this studymoderate and review, to pressure an mantle on examination of thework the silica silicate impact polymorphs beryl. of Other andof than the few these frame- silicates silicates, to themized our structures knowledge as have been abond geometry-opti- function length and ofThe angle pressure reader data for are istherein which for available referred structures, experimental for equations toties comparison. of generated [115–117] state for mantle and and related and lower proper- mantle the materials. tion references of stable andundertaken by metastable Boisen structure typesdom for arrangement silica was agreement of of Si the and Si– O atoms. Given the close the structures ofvation even the of simplest theminerals crystals’ as with structure MgO theTo types and deri- this NaCl for remaining end,fold: such beyond (1) the relatively our the goal simple grasp. of Hawthorne’s challenge is two angles observed fordisilyl oxide quartz molecule and (Fig.ergy 3a), those function an was analytical calculatedits potential constructed for geometry en- for optimized the thesurface structure molecule displayed and based the innumber on potential Fig. of 3a. energy annealing calculations With and was the quasi-Newtonwith completed, function, strategies. either using a Each three simulated large or was six started SiO which were randomlygeometry distributed with in acalculations unit yielded cell several of hundrederate variable relatively low energy to model mod- lite structures, and including mixedcristobalite quartz, layer with cristoba- stacking sequencesgenerated. quartz The of being space tridymiteture group the and symmetry matches most of thatStructures the abundant were observed quartz phase generated struc- in thatto match each the or case frameworks arelite, structures closely almost cancrinite related observed exactly. and forderived NaJ-zeolite by bikitaite, in Smith soda- [120]several concert and cases, with O’Keeffe two a and translationally number Brese [121]. equivalent In unit cells of . þ 4 –O and et al . [84] A along a . On the 1 A et al ) maps [82], (Fig. 13), the r ( c L triangle for stishovite. 3 330 kJ mol ) and –O vectors and local lone r ( ) maps are also similar. h r ( r ) map. With the OH vector r D ( as determined by Pawley ) are useful tools for locating h r c ( L ) to isolate those domains in a ma- required for the incorporation of H r ) and ( r ( h h determined in an earlier study [113], the ) and OH r 6 ( h þ þ 3 Al ¼ An ELF isosurface enclosing the OSi triangle. Further, the H location is virtually the same 2 3 As observed for the two silica polymorphs coesite and O O group. The H atom is predicted to be located 0.96 3 [112]. Employing first-principles plane wavetions VASP calcula- [100–102], theclose position to of the the experimental Hmodel value atom structure. for was With an the predicted Al-bearingto H stishovite the atom defect placed AlO at a position next vector that radiatesmaxima from of the thelegend feature O for ascribed isosurface atom information to and and the the passes color lone through of pair the the domain. atoms. local See Fig. 4 those displayed by Given the efficacy of terial ascribed to bondphysical and appreciation of lone the pairparticularly function electron since is pairs, not the a trivial functiontheoretical (cf. deep is method [69]), largely used in independent its of calculation. the H atom moved upon geometrysition optimization determined close to with the the po- By the ending of yet the promising lastthe problems century, confronting derivation one mineralogists of of was the the stable most and challeng- metastable structures that Derivation and geometry optimizations of framework silicate structures at pressure þ cular to the unit cell vector Fig. 13. 16 Local bond pair maximapair occur maxima along the areSi Si– located on opposite sides of the O comprising the basis of an electrostaticobtained potential map, a Spackman similar resultOSi with H next to the O atom of the bond vectors are also siteslarity of of potential the attack. Like features the displayed simi- by sites of potential electrophilic attack,in particularly conjunction when used withoped polarized domains IR ascribed spectra. toby the the The bridging lone well-devel- O pair atomsthe electrons for nonbridging diopside, displayed O tremolite and atomscribed talc to of and the forsterite bond and pair the electrons domains displayed as- along the Si– stishovite, the in stishovite under lower mantle conditions. as that determined inStixrude an [114] independent with studyprovides first by an principles Panero underpinning and methods, for the a coupled result substitution that Si final distance from thethe O reduction atom in was energy found amounted to to be 1.02 oriented perpendicular to the unit cell vector This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , / E 17 A 2 –O –Si @ trian- 2 3 . [138] –O– R –O bond –O bond –Si angle , again in –O bond values de-  –Si. These 12.75 GPa. et al i –O bond of and 1.617 –O– ¼ , respectively, –O– P A A –O) . [138] obtained Si– at (Si–  . [137], individually, et al R ¼ff h , shows little change compared with the ex- q i –O bond lengths for compared with the ex- 5% narrower than those –Si angles for quartz et al A wider at a given pressure  and 1.613 –O)  –O– 2 A –Si angle of 145.5 [137]. In an independent opti- (Si– R  h , comprising the Si– is the average of the two bond –O– i , with increasing pressure to 18 GPa –O bond lengths generated for 276 GPa compared with an experi- and 1.614 –Si angle involved in the OSi 298 GPa. A R 0.0 GPa to 126.97 ¼ –O) A ¼ 0 –Si angle and increasing pressure. In ¼ –O– –O bond lengths, 1.611 0 K –Si angles can be expected to decrease (Si– 0.01 P K R –O– h at –Si angle is 142.2 –O–  –O– 5 N/m where –Si angles are as much as 10 GPa, using the LDA local density approximation ¼ The optimized Si– The average values for the two nonequivalent Si– The experimental force constant for the Si– 2 –O– q bulk modulus of lengths comprising the angle and The experimental angles tendbut to they track are the systematically theoretical trend, than the optimized valuesnet (Fig. 14b). atomic The charges magnitudeelectron obtained of in density the a distributionatoms virial calculated also partitioning for decrease oflength the slightly the and Si with Si– decreasing and Si– O their study of the silica polymorphs, Demuth @ gle in stishovite is verysure, rigid which and accounts largely for unaffected its by much pres- larger bulk modulus. quartz, cristobalite andthe coesite experimental were valuestypically found determined to at within ambient agreeSi– conditions with about 1% whereas the optimized relatively good agreement with the experiment. bond lengths, slightly longer Si– mization of the quartzof structure, the also within local the density framework theory, Demuth and a slightly wider Si– mental value of results indicatelengths of that quartz, cristobalitewhile under and the coesite pressure, Si– willsubstantially, change with the little the bulkca Si– of polymorphs the like compressioncompliant of cristobalite nature the closely of sili- two the connected nonequivalent angle toquartz [136]. the calculated The and Si– average cristobalite, of the perimental value of 143.5 quartz is 597lated N/m for [135] the comparedthe disilicic with bending acid 615 force molecule. N/m constant On calcu- calculated the other for hand, the Si– angle for the molecule is much smaller, 1/ geometry-optimized the structure ofto quartz for pressures up determined at ambientmized Si– conditions. In addition, the opti- observed. For example, thelengths two optimized nonequivalent Si– pressure, for are quartz 1.608 at absolute zero and zero for quartzFig. reported 14a with by respecttry-optimized to values. several pressure The together workers optimized crease with slightly, the are geome- plotted in with increasing pressure,other as hand, the demonstrated Si– below. On the perimental values of 1.607 while the experimental bondretical lengths line. scatter As about reported the by theo- Gibbs (Fig. 14b) decreasefrom 142.20 nonlinearly with increasing pressure the two nonequivalentlengths experimental both and appearpressure. optimized to The bond decrease slightly optimized with Si– increasing 2.4 and 99.8 GPa 45.0 GPa, ¼ ¼ 0 ¼ 0 0 0 0 0 K K K 4.9, ¼ 6.3, respectively. The 0 0 -point mesh [129] were ¼ K k 0 0 K 11.5 GPa and ¼ 0 0 K 9.0 and ¼ 0 0 was generated as a function of the unit ), atomic coordinates and cell dimensions O disilyl ether have been successfully used K 37.1 GPa, 2 /c , using the generalized gradient approxima- E, 2 V Si ¼ 6 C 21) (see [128] for details) and low cristobalite 0 0 6.0, 2 3 K ¼ P data sets. With the total energy data sets and the 0 0 2). The structure of coesite was optimized, using 4.8, respectively, compared with the experimental 1 K V 2 1 ¼ Given the simulated annealing generated cell dimen- 4 0 0 P 14.8 GPa and 94.3 GPa and agreement betweenbulk the moduli experimental ispressure and comparable derivatives whereas the the is,with calculated agreement as the of expected, calculatedthan the observed. substantially values In being poorer addition,for an typically optimization the of much high the smaller structure pressure polymorph stishovite generated a K ( the quartz structuretranslational symmetry were of generated,bonded silica is suggesting interactions governed that inneighbors. involving part the Indeed, by nearest the the silica and derivation structure of types next [118,lier the 119] assertion nearest large is that number consistentas the of with if bonded short-ranged the interactions and ear- short-ranged in molecular nature silica in nature. behave ofprovides Albeit a the true, basis the for bondedcules understanding like interactions of H why not gas phase only mole- Bonded interactions and crystal chemistry: a review and the same strategy,symmetry, starting ( with the[130]. observed The space structures group determined for the experimentally threerange at polymorphs of have ambient been [131–134]. pressures conditions These up structures forson serve with to a as model aof 8 structures GPa basis unit optimized for cell with at compari- volumestotal a at prescribed room energy, absolute set zero.cell temperature For volume, each model,tion the to theparameters for quantum the mechanicalting equation of energy a state functional.E were Birch-Murnaghan The obtained by expression fit- to the calculated sions and atomicthe structures coordinates for for severalmorphs stable [128] the and were silica geometry-optimized metastableodic with silica polymorphs, VASP, plane poly- a peri- wavedensity approximation code to account [100–102]tion for to the that the exchangedensity utilizes correla- total energy. of the The the local kinetic Monkhorst-Pack energy cutoff and the to advance our understandingwater of the [122–125] reaction and ofbut quartz related it with minerals also provides atan a the excellent basis glass atomic for former level, morphs understanding and of why why there silica silicathe are is [126, so electron many 127]. density poly- reaction The distributions, of successful the silica modeling forcesupport with fields of to representative the and molecules observationdivorce the also by of lends O’Keeffe-Hyde crystal [39]the chemistry that teaching the from and molecularwas a chemistry study mistake, in leaving of both mineralogy disciplines all early the poorer. last century chosen to ensurestructure convergence and of energy.assuming the The geometry-optimized the optimizations spacequartz were ( group completed symmetries observed for low values of associated set ofthermal unit cell bulk volumes, modulilated the for and quartz, zero cristobalite the pressure and iso- coesite pressure are derivatives calcu- This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder.  ). d et al. ) and , for when i a  –Si, for , the cal- –O)  [138]. The –O–  (Si– Si– ) and for the 180 in Ref. [137]. It R . c ff –Si1 angle of –O bond length, h A 180 A values calculated e –O1– ) and for coesite [130], and five nonequiva- for quartz [137] ( b P A , for the two nonequivalent i Scatter diagrams of the aver- –O1) bond length invol- –Si angles, –Si angle plotted with respect –O) or whether it appears to be (Si1– –O– –O bonds calculated for quartz at –O bonds–O– comprising a given  (Si– R R Si– The lines in theto figures were calculated splinespheres. fit data The open plottedand symbols for the as quartz represent colored black symbolslength for the coesite andSi– angle experimental data.zero bond One pressure of isTable 1 given the as incorrectlyshould 1.6804 read in 1.6084 Fig. 14. age experimental Si– h for coesite [130]quartz ( [137] ( both as a function of pressure ( Si– Si– to pressure and 1.620 –O bonded interactions ranging in G. V. Gibbs, R. T. Downs, D. F. Cox A 1 symmetry, where the constraint no b P d is assumed, the structure was geometry- /c angle decreasing the most with increasing 2  C also resulted in an angle of –Si angles that range in value between 137.5 with the shorter bonds involving the wider an- /c –O bonds of quartz at 18.6 GPa. 1  2 –O– P Levien et al. 1980 (exp.) Levien et al. 1980 (exp.) Hazen et al. 1989 (calc.) Gibbs et al. 1999 (exp.) Jorgensen 1978 (exp.) Glinneman et al. As observed above, the structure of coesite comprises The average experimental bond length, P GPa P GPa Gibbs et al. 2000 (calc.) Ross et al. 2003 (exp.) Levien and Prewitt 1981 (exp.) culations indicate thetion angle of in coesite dynamicinteractions is that disorder not determine butpendent a the is manifesta- structure. optimization a Further,group an of result inde- of the the structure bonded assuming space space group until a substantiallyresult that higher is pressure consistent is with the reached small [139], a eight nonequivalent Si– lent Si– for the Si– and 180 gles at ambientbate, conditions [134, however, 141]. as There to has been whether de- the Si1– the structure is actually 180 pressure (Fig. 14c). The latter runs contrary to the general fact that the experimentalfor thermal the displacement O1 parameters small atom determined and over virtuallyremaining a range four the of nonequivalent samefurther pressures O are as evidence atoms those in thatand not the observed indicates a structure for result that is of the the dynamic disorder angle [131, is 134]. straight straight, but it iscates actually like bent high asorder cristobalite observed with [27]. for structural As several or sili- the dynamic angle dis- is constrained to be 180 optimized assuming length between 1.595 the two nonequivalentthe bonds in five coesite nonequivalentpressure involving with angles each the of decreases shortest with increasing ving the 180 longer holds, over thethe range angle of for pressures studied these [130]. structures As remained at , the 024681012 2 0 4 8 12 16 l

144 140 136 132 128 124 180 170 160 150 140 130 120 5 O 2 Levien et al. 1980 (exp.) Levien et al. 1980 (calc.) Gibbs et al. 1999 (exp.) Glinneman et al. (exp.) Hazen et al. 1989 (exp.) Jorgensen 1978 0.0 GPa are small, 0.025 and 0.011, P GPa P GPa ¼ P 0 4 8 12 16 20 quartz coesite 0 2 4 6 8 10 12 14 –O bonds and the onset of amorphization. Single

As observed above, the ellipticity of a bonded interac- 1.65 1.64 1.63 1.62 1.61 1.60 1.59 1.58 1.57 1.56 1.55 1.62 1.61 1.60 1.59 1.58 1.57 1.56 1.55

–O bonds at –O bond in triclinic CaSi greater the ellipticity of[2], the bonds interaction. with Asto unusually observed be large above relatively ellipticities unstablestructure are and of considered susceptible quartz to isat rupture. believed high to As adopt the pressure,calculations an it amorphous indicate state is aof of its connection interest Si– between tocrystal the explore experiments ellipticity whether completed15 the GPa for revealed quartz featurestural at that instabilities have associated pressures beena with to ascribed a crystalline to gradual to struc- values, transition an from however, amorphousSi– state calculated [139]. for The ellipticity the two nonequivalent tion has beenelectron used density as a distributionperpendicular measure to measured of the in bond thethe path a anisotropy bond, at the cross of the greater the bond section the critical difference point between of and found that thetional calculated unit coordinates cell dimensions,match of the the frac- experimental the values within atoms a few and percent. the molar volume 18 [140], the ellipticitiessmall. for Despite thequartz that bonds increase with for thesity increasing quartz ellipticities pressure, distribution are the remains forindicating electron very nearly den- that both circularrange the bonds in of bonds cross pressures in ing considered. section, are of A the relatively weakening Ramanfraction and stable and spectra broaden- powder over energy observedever, dispersive the in that X-ray the a dif- onset later of study the indicates, amorphous state, how- may not occur with the shortertion. bond As being thecities pressure more is elliptical of increased in the0.044 to cross and 18.6 bonds 0.019, sec- GPa, respectively. increaseSi– the Compared ellipti- slightly, with the doubling unstable in value, This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. A 19 1% –Si 0.01 –O– –O and –Si angle –O– –Si angle for –Si angle but –O, Al– –O– –O– –Si angle displays a –Si angle are plotted –Si angle is largely in- –O– –O– –O– –O bond lengths for thermal and the Si– i whereas more than twice that  –O) 15 (Si– –O bond lengths. R h without disrupting the structure and dis-  3 GPa is required to reduce the Si– –Si angle is substantially greater than that ob- 7 GPa) is required to reduce the angle a com- –Si angle in cristobalite is free to change from –O– –O– –Si angles (Fig. 15c) are largely invariant [145– –Si angles and largely independent of the stiff to 109.47  –O bond lengths and the Si– The values of Unlike the silica polymorphs with 4-coordinate Si, the Recently, the structure of beryl was carefully geometry- –O bond lengths observed for stishovite–O– with 6-coordi- –O– –O bond lengths, the volume compressibility of stisho- pressure ( the stability fieldtwo of polymorphs. keatitecristobalite The is shows determination that intermediate itschanged of between with bond the increasing lengths the pressureof remain structure while its largely the Si– of un- angularserved change for eitherrange quartz [133]. or In coesitethe over addition, Si– the as same observed180 pressure by O’Keeffe [144], angle in cristobalite torting its silicatechange in tetrahedra. the Aspolymorphs angle such, relative may to be a thatin expected relatively displayed to pressure large by occur without theDowns for disrupting other a and given thestudy change Palmer structure of [133] substantially. cristobalitesignificantly observed larger that angular in the decreasepressure for a Si– a than given high increasepressure that in pressure of observed for quartz. For example, a shorter thanbond that is observed inAs close while observed, agreement the the with calculated shorter the Si– experimental equatorial values. parable amount instructure is quartz, disrupted indicating towith that a increasing lesser pressure. the degree cristobalite than that ofcristobalite quartz geometry optimizedlished) at Virginia areFig. Tech 15b, (unpub- plotted respectively. Theat value against of room the pressure pressure experimentalthe is angle experimental in angles comparableculated depart with Fig. trend progressively 15a that with from calculated, increasingcreases, the and pressure. cal- but the As thechanged calculated pressure as bond in- observed,longer, but length they on are average, remain systematically rection than largely of those the un- observed. experimentalmotion Si– However, yielded a bond cor- almost lengths exactly. In that contrast,narrows match the those substantially calculated calculated served. Si– with increasing pressure,Si– as ob- nate SiSi– shorten148]. with The geometry-optimizedwith bond increasing increasing lengths pressure both pressure but decrease the while longer axial bond the is dependent of thebalite pressure and (Fig. quartz 15d).largely for Thus, which unlike dependent the cristo- Si– volume compressibility on is Si– thevite compliant is conversely nature independent of the of Si– the dependent on the Si– optimized using[150]. CRYSTAL98 In [149] the study,corresponding at the ground energy the state wavecell were LDA function volumes. and calculated level The the for resulting different optimized Si– unit Be– together with those determined experimentally by Hazen . et al 0.04. ), for each c -values gen- r j ( r -values, and their repro- e about the trends  values for each of A i –O) 0.005 Æ (Si– –O bonded interactions in –O bond increases slightly angle in coesite, the average R h Si– Si–  VI IV –Si angles measured at ambient –H interactions was asserted to in- –O bond also increases slightly with –O– Si– IV . [138] LDA calculations for coesite reproduced –H interactions increases. On the basis of the –Si angles decreases with increasing pressure and et al In a theoretical study of the effect of pressure on Likewise, the accumulation of electron density in the In their study of the silica polymorphs, Demuth –O– -values for the remaining bonds increase slightly with in- bonded interactions, the[142], properties of generated themethods with electron for density first astructed linear principles vice chain clamped molecular ofthat at pressures H with orbital up atoms increasing to in pressure 60 an the GPa, value argon revealed of con- duction of howcomparable the with cell the experimental dimensions dimensions. vary with pressure is of the H– negative value ofbelow) the together local with electronic itswith progressive energy increasing increase density pressure inof and (see magnitude electron the density progressive betweenlent accumulation the character H of atoms, the the H– shared cova- internuclear regions between Sicoesite and increases O forsmall with both range quartz of pressure and pressurescharacter by studied, of indicating the about that the 2% shared over the crease with increasing pressure. belief that acrease shorter less bond with isrevealed increasing a pressure by stiffer than bond the a and figure, longer should the one. de- As Bonded interactions and crystal chemistry: a review pressure. Also, withsity increasing pressure, is the slightlyward electron but the den- Si progressivelyenergy and locally O density concentrated atoms, ofslightly indicating to- with the that confining the bonded pressure.bonds local interactions With involved kinetic the also in exception the increases of the 180 with increasing pressure.of In the a stishovite polymorph carefulcluded of high on silica, pressure the Yamanaka basis [143] study con- of the smaller monopole erated for theenergy synchrotron Si single andshared crystal O diffraction character atoms data of in that the a the refinement of high Also, the net atomiclated charges for for both the quartznitude Si and and with O coesite increasing atoms decreaseshared calcu- pressure, slightly character in further mag- of evidence that the the e creasing pressure withthe the narrower bonded angles interactions displaying involved the in largest [138] also geometry-optimizedkeatite the structures and ofquartz, tridymite, stishovite cristobalite ineach and of addition coesite. the polymorphs toter The were than generated the molar one withcreasing percent. volumes structures accuracy pressure The bet- for for decrease of keatiteate in was between the observed that volume to of with be cristobalite intermedi- in- and quartz, suggesting that stishovite also increases slightly with pressure. the five angles scatter within conditions with a maximum error of 0.7 tracks the calculated trendgles again tending with the tothat experimental be calculated an- smaller, formuth as a observed giventhe for pressure experimental quartz, (Fig. 14d). than Si– The De- defined by the[130]. bond With lengths exceptionSi– for of the the straight optimized angle, structure each of the This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. –Si )M ) vs. et al. a d –O– –Si angle –Si, for –O2 bond ) and the c –Si angle, –O– –O– for cristobalite Si– –O– ff –O bond length, P –O bond lengths ( ) and stishovite ( b , for the two equivalent i Scatter diagrams of the aver- ) and stishovite ( The M– a –Si angles, –O2 data are plotted as blue cir- –O) –Si angle anchored at a sort –O2 data are plotted as green circles, –O– (Si– Si, Be and Al, and the Si– . [138] calculated the total en- ) calculated for beryl plotted with re- R b –O– Si– Fig. 15. age experimental Si– h bonds comprising a given Si– cristobalite ( pressure. The lines inspline the fit figures to were black calculated spheres. data plotted Thecristobalite as open symbols andthe for experimental stishovite bondgle length represent data. and an- angle vs.[133] pressure ( Fig. 16. ¼ ( length data are plotted asBe– orange circles, the Si– cles and the twosets SiO2 are bond plotted as length red and data black circles. spect to pressure [150]. Thetal experimen- values arewith plotted error as spheres barsvalues in whereas are (a) the plotted as calculated and open (b). circles The in experimental Al– (a) et al G. V. Gibbs, R. T. Downs, D. F. Cox b b d (1994) Palmer Gordian knot. In a clarification of the nature of 3 Calc. Ross et al. (1999) Yamanaka et al. (2002) calc Downs and The overall agreement between the calculated and the PGPa PGPa PGPa the high-low phaseand transformation tridymite, for Demuth ergies quartz, as cristobalite athat function the of equationsdetermined the of values within unit state thethe cell agree experimental case volumes uncertainties. with of In and thesumed quartz, found to experimentally for indicate example, thatto it the high transforms calculations quartz. continuously were from as- low experimental structures,their together electron with densitycates distributions the that first discussed modelingour principles below, understanding methods of indi- of are theparticularly destined bonded those to interactions that improve forand form minerals, low crystals quality, resistant zoned, to disordered experimental methods. Also, the highly compressible structurescristobalite like (with quartznected four-coordinated and with Si) especially thewhereas compliant are the nature directly highly ofwith incompressible con- the six-coordinated structure Si– Sivariant of is character directly stishovite of connected the Si– with the in- of OSi 0246 0 5 10 15 20 25 30 -0.50.00.51.01.52.02.53.03.5

146 144 142 140 138 136 134 132 130

1.64 1.62 1.60 1.58 1.56

1.90 1.85 1.80 1.75 1.70 1.65 1.60 1.55 1.84 1.82 1.80 1.78 1.76 1.74 1.72 1.70

Å Summarizing, the framework structures of the silica (-)Å R(M-O) R(Si-O)Å 0.05% wider than those calculated. Actually, the zero and Finger [151] inagreement Fig. between 16a the withobserved optimized respect is bond to good. pressure. lengthsquartz The As and and observed those cristobalite, for the the silica experimentalpressure polymorphs angles experimental arecounted only Si– because, unlikeoutside the other the angles,slightly it diamond was larger cell measured thanthis where those angle, determined the the indo angles the Si– cell. in tend Ignoring (Fig. to 16b). the The be theoretical silica bulk modulus, polymorphs with increasing pressure 20 reported for beryl is polymorphs and berylsures generated and at at zero absoluteods zero and described with here higher firstcuracy and pres- principles of elsewhere LDA those [116, meth- connection 138] determined has rival experimentally. been the Also,perimental made ac- a compressibilities between of direct the thetions calculated structures of and and the ex- the bondpressure. varia- lengths and For angles example, of the the structures with bulk moduli calculated for This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 4 b 21 –O –O and CO, -NaMgPO 2 a ). Two quasi- c r ( r . In later studies, 2 3 CO, CO r O 2 2 –O), for the silicates, . [140]. The calculations (M– R et al COH, H 3 O, H 2 O, BeO and B 2 C 6 –O. These results were used to explore ) increases as the electronegativity differ- ,H –O bonded interactions were extended by c 7 the calculations yielded the bond critical r O, Na ( a 2 O , r 2 –O and for second row bonded interactions C –O bond lengths, 6 –O bond lengths and the electronegativity differ- , between the bonded atoms tends to decrease. c ,H D 4 ) and in Fig. 17b with respect to –O through S– The properties of the electron density distributions cal- c The M– CO r 4 ( were completed for thelite, silica coesite polymorphs and quartz,low stishovite, cristoba- cordierite, the danburite, framework lowthe albite, silicates, chain maximum beryl, microcline, silicatesmene, tremolite, the diopside, sheetite, jadeite silicate fayalite, and datolite, tephroite, spodu- the topazperovskite, orthosilicates and wadsleyite chrysoberyl and forster- together thebromellite, oxides with Li corundum, periclase, and the local energysity density properties distributions of theatoms together electron were den- calculated within with the principle TOPOND is [156], net the softwareerties same for that charges the as electron of that density used distribution the to for calculateculated molecules. the for prop- the bulkstudy of together the minerals withreported [54] in considered their Table in 1 experimental this of bond Downs lengths are ence, additional calculations wereother silicates. completed Also forcalculated included for a in the the number phosphates study AlPO4–15 of were and properties and the sulfatesthe anhydride first and row vanthoffite.including M– The properties trends that for wereH calculated for the molecules each geometry-optimized atvel. the In BLYP/6–311++(2d,p)point short le- and localatoms energy density bonded properties to for O first row for M the bonded interactions Li– through C– Na– the variation inas the the properties M– ences of between the the bondedright M interactions in atoms the periodic and table. O decrease from left to parallel trendsbonded are interactions displayedthrough involving in C the andsecond Fig. 17a, the row first second one atomsthe row for Na value for of interactions through atoms the involving S. Li the Along the two trends, phosphates, sulfates,bearing oxides, molecules carbonates arer plotted and in the Fig. 17a C– with respect to ) r ( r D ) b þþ –O bond ) and ( c r ( r –O data are for , for a variety of ) A a –O bonded interactions –O) (M– Experimental M– R ) values. The blue open sym- c ) level. The bond critical point r p ( , r –O interactions involving the sec- –O bonded interactions in calcite d 2 properties for theculated molecules with were XTREM, cal- softwareprovided by kindly Richard Bader. r bols denote M– molecules and theoptimized bond at lengths(2 the were BLYP/6–311G involving the firstBe, B row and MM– C atoms and the Li, ond red row ones M denote and atoms Na, S. Mg,C– With Al, Si, theand P magnesite, exceptions the of C– the lengths, Fig. 17. Prior to theCrystals formulation (AMC)’ of theory the fortions characterizing ‘Atoms in bonded in termsand interac- Molecules of his and colleagues electrondirect [2, density 152, aspherical distributions 153] byfield and atom and the Bader modeling the strategies Laplacianmental for of of the and a the multipoleStewart theoretical modeling gradient [154], of electron vector experi- X-raymarily density diffraction to distributions methods identifycrystal were by crystalline structures used materials pri- and, and to in determine fewer cases, generate Calculated and experimental physicalties proper- for model electron density distributions calculations forphates, different sulfates, mineral carbonates andsilicates groups arsenates, and as such done belowprovide as here a for for phos- more the generalof sulfides, improvement their properties in can and our the be understanding crystal expected chemistry of to minerals. Bonded interactions and crystal chemistry: a review maps. During thedensity latter distributions, part thebutions physical of and properties the the local ofmined last energy the decade, density with distri- properties electron minerals were relatively and deter- high silicicperimental acid accuracy electron molecules density forproperties [54]. distributions have a In been and contrast, recordeda number the with few ex- physical such minerals. of accuracy Intheoretical for a model only recent electron surveyported density of that the distributions, experimental the itare and physical was comparable properties re- with ofnumber those the of determined electron cases experimentally density high when in generated a energy with synchrotrondata high resolution single [155]. and crystal Insomewhat poorer contrast, X-ray in the diffraction tributions several cases agreement for was recorded electronsources found density of dis- with toelectron X-rays. be density conventional In distributionsTAL98 this were [149], lower generated study, using withcell the the energy CRYS- dimensions experimental model andon space theoretical the a group coordinates modeling types, a of of the the linear periodic atoms. combinationthe wave Based program of functions generates in Gaussian theoreticaltions terms electron based at of density the atomic distribu- Kohn-Sham orbitals, and the LDA levels. The physical Earth materials plottedthe with respect to theoretical ( This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , ), c A in- –O r ( et al. 3 r along Si– l 2 1.47 IV A r . Rather value, the A c 1.23 –O bond, the D , and –O) 3 2.44 are substantially l B– is not only posi- –O bond vector, 2 IV 3 ( l l R þ –O) ) with decreasing bond , measured perpendicu- and c j 2 , [159] are each bonded r 2 l 8 1 ( Ca– l l r O 2 2 þ VII þ ( 1 along the B– r Si R 2 1 l l –O bond vectors, a feature that j –O bonded interactions indicates A ¼ 2 = ) 1 G. V. Gibbs, R. T. Downs, D. F. Cox c and –O bond vector and r ( 1.00 –O bonded interactions, are plotted in –O). As the bond length decreases, r j¼ 2 A 1,2 (Si– r l R , j and hence in 3 1.62 l 3 ), ) values, the magnitude of the average curva- c l c r ( r along the Si– ( r –O bond vector. This result substantiates the r A –O) increases at a substantially slower rate than –O bond critical point physical properties determined Si– j ). With the possible exception of the C– As observed above, an individual O atom for a given c The experimental electron density distributions and the Clearly, as displayed by the electron density distribu- The –O interactions are typically straight rather than curved, IV r 1,2 0.94 Si– ( ( l R with high energytion synchrotron data single for crystal stishovite X-ray (blue diffrac- spheres) [94] and the smaller than VI M– closely paralleling theindicates M– that the bonds are largely straincrystal free [2]. often displaysbonded a radii, relatively particularly when wideeral the different range atom metal of is atoms.O bonded different For atoms to example, sev- in the danburite, nonequivalent CaB tures of length for each ofthat the electron M– densitythe is bonded systematically atoms,the bonded contracted progressively atoms [4]. shielding toward the nuclei of calculated for the Si– Fig. 18 (red openbond squares) with lengths, respect toj the experimental to three different metalexperimental atoms, B, Si bond and Ca, with lengths average of than abonded single radius bonded of radius, each O atom displays a creases. As the magnitudes of tive, but itwith also decreasing increases bondr relatively length rapidly and in the magnitude increasing value of increase in properties determined for scolecite (orange spheres) [160], lar to the bond path, the third curvature, the Ca– O’Keeffe-Hyde [39] claimatom like that thesistent the O with atom radius the isO of not observation atom single that a correlates valued. the withtween bonded It bonded the the is radius electronegativity M also of difference and con- be- the O atoms, the greater the larger the radius of the O atom. tion observed foran danburite O and atom, otherthe as silicates, measured the kind along sizereaching of a of the bond atom O vector, atom,tion, and depends the on the longer the the largerclear lengths bond the that in of the bonded a bondedalong given the radius radius direc- a in bond of the that bondother vectors O directions. direction. vector atom Further, is It andsity on only is the distribution, defined that basis the itas of O the is rigid atoms electron spheres in undefinedpolarized, den- danburite each depending along do on with all which not the a it behave properties well-defined is ofradius radius bonded. the of In but M an point are atoms Olengths of to atom fact, is rather the furtherbonded variability than justification interaction of for in the radii usingother a bond or in structurebased whether can determining on be one its whether position replaced structure in by a one is an- structural field favored map. over another , ), Þ to c c r r (M) –O) ( ð ) in- –O), b A c r r r r –O) is ( (B) in- (M– (M– –O) in- –O) de- b r (O). As . In con- R ) is posi- r (M) for a R 8 b c (M– b r r O , for a given r ( R (M– (Al– 3 ) for each of ), the expec- r A R R c ) results in a c 2 c r A ), r –O) and ( ( r c ) and r ( r r r c 1.5 ( r r 0.1 ( r (M– (M) also increases b r –O bonded interac- R (Al) increases from 2 r b and as r to r , ), A A c A r ( –O pair of bonded atoms, ) adopts a arrangement such 1.1 r r 2.1 ( when bonded to an electropo- 1.35 ), the larger the value of –O bonded interactions. In ad- r c r A ( to (O) for the M– to –O) for first and second row M r b –O) increases, r 1.5 A . Overall, the increase of A (M– ) also increases for each of the bonded þ A R (M– c r to –O bonded interactions, (O) increases linearly with increasing R ( b 0.87 1.70 3 r r 2 (M) 1.25 b r r when it is bonded to an electronegative atom to A ) are more meaningful in the sense that they in- as its coordination number increases from 2 to 8 c ) increases linearly with increasing A r (O), respectively. The bonded radius of the O atom ( c b 0.8 r –O) –O) along parallel trends with the radii of the O A r r –O) increases from ( The bonded radii for a M– r 2 0.95 (M– (B– (M– creases with increasing 1.28 [30]. Like the parallel trends between the bonded interactions.oped A trend similar exists but fordition more the to poorly C– devel- being highly correlated with the shorter the[2] bond that in an agreement equilibrium with bond Bader’s length assertion decreases as tive and relativelynegative large, particularly atoms for likedisplayed the Si, in more B, Fig. electro- 17 P between and S. Inherit in the trends for eachR of the bonded interactions. For example, as r interactions with decreasingtion bond of length. several With C– the excep- and substantially greater decrease inbonds bond than length that for displayedby the for Fig. longer 17b, the shorter ones. As evinced like C in CaCO displays a relativelyinteractions wide displayed range in ofof Fig. values 17, for ranging the between bonded a value R atoms bonded tobonded second to the row first M row atoms M exceeding atoms those by Also, for each of the bonded interactions, 22 given increase in atoms is smaller than that displayed by creases from R the value of tions used to construct Fig. 17, the bond paths for the sitive atom like Na [54] in low albite, NaAlSi trast, the ionictively radius small of range the of oxide values, anion varying displays a between rela- 1.21 creases. In general, a given increase in defined by Bader [2]M as and the O distance between and the the nuclei bond of critical point, are denoted that the energy of the resulting configuration is minimized. volve well-defined observablestrength-, quantities valence-bond length unlike correlations.tion the The of bond accumula- theassociated electron density with along amirrored a by lowering bond a path ofpair virial is the along path, not which energy, only a thestabilizing line but potential [158]. that energy it The density connects accumulationalong is is the of maximally the also bonded the electron bondpected density paths to of stabilizebond the paths a [69] bonding structure given interactions that relative is to ex- one that lacks and tation value ofpoint the [157]. density As operator such, at the correlations the between bond critical bond length. As creases from also highly correlated, asvalence discussed and bond above, strength. withnot These both quantum parameters, bond mechanical however, observables are like This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . b d 23 et al ) values c r ( r ) were reported. The c r ( r 2 ) values fall slightly outside the calcu- c r r ( r 2 and r a c Þ c r ð r Despite the comparable calculated trends, Gibbs –O bonds for these silicates are plotted in Fig. 19 atoms serves totheir shield separation the decreases. nuclei, as discussed above, as for danburite falltrend well but within the the scatter of the calculated lated trend withagreement of smaller the experimental danburiteis data values. a with The the testament[159] calculated in close of collecting values the the data care set. In used contrast, the by experimen- Downs and Swope [155] found thatmodel the experimental physicalated electron with properties density conventional determinedexperiments distributions lower for gener- often energy X-raytrends. depart single somewhat For crystal Si– from comparison, the thewhere calculated experimental it datarange is displayed for apparent byscatter that the somewhat the a outside calculatedburite the number trends [159], trends. scatters topaz For whereas [166]ues the within others and for silicates diopside the dan- [103], only the val- –O bond. [162], te- 4 SiO 2 olivine [164] struc- 4 SiO 2 3 l ), b [162], ( j 2 –O bond l 4 + [164], green [161], fayalite, Fe SiO 1 4 2 l j 4 2 [163] and Co = SiO 1 2 4 ) (red open squares). SiO c 2 j¼ r ( SiO 1,2 r 2 2 l (open diamond) [104] j –O), for the silicates dis- [161], Fe r ), [163], Co ) 6 (Si– a The figures on the left and The experimental Si– d 4 4 R O 2 )( c r SiO SiO ( 2 2 r ) and ( c Superimposed upon the calculatedare experimental data bond lengthel and mod- experimentalproperties measured bond withsynchrotron critical high and high energy point crystal resolution single datacoesite, (stishovite, black blue spheresMg and spheres; the olivines tures (green spheres) [165]gle and crystal for data theare high collected resolution plotted for sin- in coesitebond Fig. lengths (black 18 for spheres)calculated with comparison. [86] and The respect the agreement to experimentalseveral between data cases, their the sets demonstrating experimental isto that comparable a the stage in theory whereto has the those calculated advanced determined propertiesdecrease, are experimentally. there often As similar isdensity the a at bond progressive the lengths surface accumulation bond associated of critical with the electron tron points local density and concentration toward at oftions the the the are elec- bond dominated interatomic path. bydensity along a However, the larger the bond locallocal interac- path, depletion electronic resulting of energy in electron densitytant an together increase increase in with in the The the the progressive concomi- shared local interaction concentrationparallel for of the to the Si– the electron density bond path in the direction of the bonded Fig. 19. right arerespectively, copies with of dataconventional Fig. recorded 18a lowercrystal with and data intensity for d, danburitediopside (open (open single circle), triangle), datolitesquare), (open LiGaSi topaz (open star) and forsterite, Mg ( Fig. 18. length, played in Fig.to 17 plotted with respect Bonded interactions and crystal chemistry: a review Mn superimposed on the figure. spheres and scolecite,[160]. orange spheres phroite, Mn This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) c r ( –O –O), et al. r ) and c r ( (Mg– ) values c r R r –O data are ( r ) and –O bond lengths, c r ( ) (Fig. 17a). Also as c r r 2 ) (bottom). The experimen- ( c r Experimental and calculated r r ( r –O data obtained with synchro- ) (top) and experimental and 2 c –O), plotted with respect to r r ( –O bond lengths, r 2 (Fe– plotted with respect to tal Mg– tron radiationspheres are andplotted plotted the as open as calculated greenperimental squares. synchrotron green data The Fe– ex- plotted are as bluelated spheres data and are plotted the as calcu- blue squares. r Mg– Fig. 20. and calculatedR Fe– –O bonded interactions for danbur- G. V. Gibbs, R. T. Downs, D. F. Cox , of the atoms comprising the interac- r –O and Na– As observed above, the first and second row M– –O bond lengths decrease and scatter along quasi-paral- –O bonded interactions are divided by the periodic ta- atom like Sb. The experimental for the Ca– ite and the fibrouscalculated zeolites [155]. are Overall, alsoculated the comparable properties agreement with and those betweenand those the synchrotron obtained cal- single withcates crystal high diffraction resolution is data comparable.experimental for In the data sili- contrast, collectedmethods the ranges with from agreement conventional moderatelyment good between diffraction between to good. thehigh The calculated resolution agree- and experimental the propertiesassertion supports synchrotron that the and the earlier accuracy the calculated with properties thetions are experimental can comparable data be in andthe undertaken that generation for the ofdeed, calcula- a useful wide a crystal range recent chemicalproperties of for information. study the minerals In- of Nidemonstrated in sulfide the that heazlewoodite the (see bondelectron calculated below) critical has physical density point properties distributionsthose of for physical the the observed. sulfide Itwill closely match is improve anticipated ourtions for that understanding minerals future ofcrystallized, like twinned the calculations the and sulfides disordered. bonding which are interac- oftenbond poorly length datatrends decrease and when scatter plottedvalence along against (Figs. quasi-parallel 1a theM– and bond 2a) strength inlel and much trends the bond same when way plotted that against the observed above, when theM– individual bond valencesble for the row number, tions, the data scatter roughly along a single trend (Figs. 1b ) c r –O –O ( r )va- c r ( r 2 –O interac- r ) values that c r –O and Fe– ( r –O bonded inter- 2 r –O bonded interac- –O interactions for te- –O bonded interactions for –O and Fe– , datolite, diopside and topaz are 6 ) values are smaller in value than O c 2 r ( r 2 ) values each increase, the experimental c r –O bonded interactions [165]. Model phy- r ( r olivine [164]. In Fig. 20, the Mg– 4 ) and , are also comparable with modeled experimental c 3 r ( –O and Co– SiO O For bonded interactions other than Si– r 2 2 –O bonded interactions for stishovite and coesite, the –O bonds. Similar agreement exists between the experi- 2 experimental physicaldistributions properties for of the the Mg– electron density bond critical propertieslite calculated and for several forsterite otherresponding minerals and are experimental faya- comparedSi– with values. the As cor- observed for the tions scatter within orcalculated close to data. ther scatter displayed As by the the experimental and calculated sical properties recentlySb calculated for senarmonite [155], mental andMn– the calculated physical properties for the bond lengthsSi– decrease nonlinearly as displayed by the in somewhat poorer agreement with the trends with values that are uniformly larger and tal values for, LiGaSi 24 properties [167] obtainedfraction with data, high particularly resolution given X-ray that dif- it contains a heavy lues with lowgiven energy that single-crystal it data is is a second understandable derivative. actions, high energyfraction synchrotron single data crystalbonded have X-ray interactions dif- for also forsteritetions been [161], for the recorded Fe–phroite fayalite for [162], [163] theCo the and Mg– Mn– the Co– the calculated onessite. and The the difficulty in experimental determining values comparable for coe- are smaller thanexceptions, the the calculated values. Also, with a few This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , Þ 4 c 25 c r for ð r SiO 2 similar 4.5 0.19 ) r )/ c are defined, the r –O) trends. The ( Þ r e –O)/1.625) c ) values determined r c (M– ð r R ( (Si– r r R line indicates that there 1.42( (  vs. ¼ ¼ and s s –O) –O bonded interaction, the great- –O bonded interactions used to construct (M– R –O bond lengths used to construct are connected, but it also provides a r )/ , the greater the accumulation of the elec- c b s r ( r and r –O bonded interactions are plotted against the / s agreement between thethat two expressions not only indicates to that used to model the the experimentalFig. Si– 17 [169].the The Si– resulting empirical bond valences for correlation between thesimilar properties two of indicates aer Si– the that value they of tron measure density atthe the bond. bond As observed critical above, points not and only the can stronger the bond va- power law expression are plotted in the figure using the is an one-to-oneempirical correspondence, on bond average,density between strength the at andstrengths, the the calculated bond value withfor critical of the stishovite, point. experimental the forsterite, The bond fayalite, electron empirical lengths tephroite bond and Co experimentally with synchrotrondata single [94, crystalpoints, 165]. diffraction the With bulk ofclose the the to exception experimental the valuesdifferent of track scatter bases the within of or upon olivine the which data calculated values. Despite the values for thedata track interactions roughly in along Fig. the 21. 45 The fact that the theoretical basis forbeen established the between empiricalbond bond correlations number length, that [50,tion, bond have the 52]. valence empirical In and bondBrown-Shannon an valences were expression illustration calculated of with s the the connec- d ¼ –O), –O) . When (M– r R (M– R ) values used c –O), and the bond critical point properties of the Si– r ( r (Si– a R –O bonded interactions the , [53] s . Symbols are defined in Fig. 18 4.5 –O bond lengths, ) values for the Si– calculated with the Brown-Shannon [50] c r s ( values were plotted against –O) and r r / –O)/1.625) Þ (M– c R r (Si– ð R r ( . As the trend in Fig. 17a is similar to that ¼ 0.22 s ) The calculated The experimental Si– r / s Fig. 17 (redcalculated open for squares). a variety Superimposed of on hydroxyacid molecules the [171]. figure (blue spheres) are the optimized bond length and bond critical point property data to prepare thethe Fig. 17a resulting were likewise divided by used to prepare Fig.empirical 17 bond (red open valence squares) plotted with respect to the expression Fig. 22. 1.39( and 2b) that was modeled with the expression Fig. 21. Bonded interactions and crystal chemistry: a review found between legend. the data[168]. also A regression scattered analysis of roughly the data along set resulted a in the single trend This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. > –O –O )is j et al. c ) r r ) and ( ( c r V H ( j r , –O, Na– by the expres- . [172]. (For i i.e. n ) are determined, r –O), for the M– ( et al ) increases while ) is negative, the r c c 2 r r (M– –O bonded interac- –O, Be– ( ( r R increases. In reality, G H ) is positive. For the –O bonded interaction c r ) and decreasing bond ) summed over all of the Þ increases whereas when ( r c c ( ) and r i r H Þ r ( ð r ( c ), and the electronic energy den- r ) is negative ( r r c G )]/ c r ð –O and S– r r ( ( ( r i V r H Ár . [174], Macchi and Sironi [175], ) –O, P– G. V. Gibbs, R. T. Downs, D. F. Cox r ) and bond length, ( i c ) in au [3]. r r et al r ( and their occupation numbers ( ) and the local energy density properties r V i [ c G i r r 2 n –O, Si– þ ( ) r . [176], for example). þ i c r P ) ( ), 8 r ) is negative for each of the bonded interactions c ( = G 1 c ) is not only negative, but it decreases linearly r et al V ( c r ( ¼ r –O, Al– ¼ –O interactions where ¼ H ( ) ) The connection between the local kinetic energy density, H ) c r ) can be generated with the inner products of the gradients of H r r ( ), r ( ( and a decrease in bond length, ( r G r )). ) is positive, the) kinetic energy increases density in dominants and value as ( H ) decreases such that ) decreases in value as 2 ), the local potential energy, G c ´n [73], the second by Cremer and Kraka [41] and the c c c c c Indeed, the classification of a M– c r ) can be determined with the local virial expression r r r r r r –O, C– r r 2 r ( ( ( ( ( ( 4 ( = Sham-Kohn natural orbitals. Once the orbital densities at the bonddisplayed in critical Fig. points 23construct for of Fig. the 17. the minerals With and bondedat an molecules accumulation used interactions, of to electron is density r V sion B– tions, increasing bond length. Thus, when G Fig. 23. G 1 V H length. In contrast, for the remaining interactions, displayed in Fig. 17 except for the Li– and Mg– with the increasing value of sity, have also played athe central classification of role bonded to interactions. one degree oris another at in the hearthis of coworkers the (cf. AMC [69]).gies theory On have forged the been by basis proposed Badernumber of for [2] the being classifying and theory, interactions satisfactorythese, strate- with three to a of one thewill more degree be highly considered. cited orEsse and The another. simpler first strategies Of wasthird proposed was by recently Bader proposed by and Espinosa bonded interactions used tosity construct Fig. 17. The kinetic energy den- positive and decreases linearly with decreasing magnitude of theV potential energy density dominants and G other classifications ofGatti merit, [69], see Bianchi Boneand Stash and Bader [173], ) 0 r ( –O )is G < r ( ) G r ( O –O bond r ), regions 2 r – – ( ), evaluated r c G r ( ) and the local r V ( V þ ), ) r c ( r 0 and where 2 ( G ) is given by the gradi- –O bond lengths and G r > ( ¼ G ) ) , regions are defined where r –O), j c ( ) r ) is always negative. As ob- r r ( r ( 2 ( (M– H V V r j R ) (in atomic units, au) where r ( ) is locally concentrated and the po- G r ( 2 ), and the local potential energy den- –O bond surprisingly well. It also pro- r r ( is necessarily greater than 2 þ G j ) ) . [171] are compared with those calculated r r ( ( V V ) can be evaluated using the expression for the j r et al ¼ ( for the Si– ) ´n [73] observed that the Laplacian of the elec- V ), by means of the local form of the virial theorem )). By mapping those regions where r Þ r ( r –Si angles observed for small silicic acid molecules ( c ), ( r r r V 2 ð ( G As discussed above, the Si– ) is locally depleted and the kinetic energy dominates. r r –O– r r 2 4 ( = are virtuallypolymorphs. the Also, same thetively as bond large lengths those number optimized ofcal observed for molecules properties for in a correlate much withFig. rela- the 18 the the same silica physi- for waybetween as silicate the those crystals. displayed bondcules in To critical and establish point the the silicate properties connection crystals, for the the optimized mole- Si– vides a basisful for why in bond modeling[44]) strength and has the glasses been (cf. bonded so [170]). interactions success- inSi– crystals (cf. lengths and the bondculated critical for point the physical moleculesby properties (plotted Gibbs cal- as blue spheres) studied local virial theoremfor (see Fig. 23 legend for an expression sity, 1 In a carefuland Esse consideration oftron density atomic distribution interactions,energy is density, Bader connected to the local kinetic Classification of nontransition metal M lence be usednumber to of reproduce materials,of the but it bond also lengths closely for mimics a the large value 26 necessarily greater than r The connection between bonded interactions forsame. the The two agreementcritical systems between point the are properties bond virtuallysilicate lengths for the molecules and the provides bond silicastanding an polymorphs why additional and gas basisbeen the phase for molecules successfully under- likesciences, used disilyl advancing as oxide our have modelthe understanding silica structures polymorphs, of for inmolecules the example, at the reaction with the atomic water of geo- vides level. and a The related agreement basis,the likewise as pro- force discussed field above,successfully for for used a understanding in tinylarge why molecule the number disilyl generation ofbelow), oxide of structure including has quartz the types been and structure for cristobalite crystalline of [118]. silica a (see for the silicates inlarge Fig. part, 22 the (plotted datato as for open the the red trend squares). molecules defineddence In track by that within the the or silicate bond close crystals, critical additional point evi- properties and the Si– is positive definite and tential energymapping dominates regions in where magnitude. In contrast, by are defined where and where electronic energy density, bonded interactions served by Parr and Wang [3], ents ofr the electron density and, with a knowledge of This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . ) 1 3 c A to to to 27 r –O –O 5 ( A and 1 –O) –O) –O) . r 3 , and A as the 1 A )/ B– A A (B– c 1 (Al– III r R ) ratio is ( ( –O inter- R c R 1.5 e r , G and as the 1.95 –O and a ( 1 to 1.595 –O bonded Si– to 0.95 e/ r Si– IV , respectively. A 3 to 1.48 Si– )/ A c to VI A to 1.36 e/ r A A ( VIII 3 to 2.32 e –O) G 1.62 –O interactions in- A 1 ) decreases linearly –O bonded interac- –O) decreases from A ratio increases. Spe- 1.74 c to 2.0 e Al– r –O interaction [183]. Þ ( Si– –O) decreases progres- VI 1 ), is asserted to be less c (Si– –O, a r ( c r IV Si– for the R r ð indicating that the Si– )/ R 1.45 ( (C– c , IV r ) increases from 6.2 e/ –O) 1 r r / 1 R –O bonded interactions, as c A ( polyhedra decreases in suc- Þ –O) r , )/ c ( 1 c Si– G 6 ) ratio, it was found that a r 1.5 e r c r ð ( Al– IV 2 r –O bonded interactions used to ( ( –O) 0.60 G IV G –O) with respect to the ratio ( R r r AlO for the B– R )/ IV c (M– IV to 2.7 e to r ( A to ( R –O triple bonded interaction for the to 2.5 e 1 R ) increased from 0.92 e/ 1 A G and –O bond increases with decreasing co- . [75] found that the c A r 5 ( ) increases from 0.50 e/ 0.15 ratio increases in value from –O bonded interaction increases progres- r c et al 1.75 r for a shared interaction and greater for a 1.0 e Þ ( 1.85 0.5 e c AlO r r V 1 ð –O interaction to an intermediate , r 4 as / . Also, as the ratios for the Si– , the coordination number of the Al atoms compris- Si– Þ 5 1 c A r A –O) –O bond length shortens from 1.96 –O) –O double bonded interaction for the carbon dioxide AlO ð VIII –O) and the Si– G VI –O bonded interaction display ratios less than 1.0 e Al– As observed above, the value of the local kinetic en- –O bond length decreased from 1.621 1.35 VI V 2.0 e C– (M– ( ( decreasing coordinationshifts number, progressively the toward thecian bond nodal from critical surface a of point distance the of Lapla- R closed-shell interaction [73].the In local density a propertiescoesite, molecular for Gibbs modeling the of bonded interactions for 45.4 e/ ratio for thesively from C– the Si– ergy density per electron, ratio is greater than 1.0, then the tion qualifies as aacter. It bonded is noteworthy interaction that of intermediate char- substantially greater than 1.0 e cession from 6 tobond 5 to lengths 4, respectively, while decrease the from interaction to IV and as suchthe classify remaining as interactions shared-interactions. are The all greater ratios than for 1.0 e ing On the basischaracter of of these the results, Si– it is evident that the shared ordination number andionic bond length from an closed-shell and the value of Si– for each ofand the as interactions demonstrated withdeceases. In below, decreasing the as bond casethe length, the of the coordination Al– number In aR more recent study of the connection between cifically, asbonded the interactions ratio from increases in value for the B– Similar trends holdinteractions for the with bulknumber the of decreasing the bond as remaining the length bonded and the coordination crease from accordingly, classify as closed-shellthe C– interactions, including molecule and thecarbon C– monoxideFig. molecule. 24a actually Despite shows these that relationships, decreases from than 1.0 e scatter diagram of action [183]. If we discount the fact that the bonded interaction isteraction closed-shell of ionic intermediate ratherfound character. than to increase Further, an linearly in- the from ratio 2.21 e was R (Fig. 24a) organizes theconstruct M– Fig. 17 intoincreasing several linearly distinct with trendsdination decreasing with bond number the length ratio [184]. and coor- One Mg– ; ) ) ) ) 3 is 3 c c r > or ( r r A ( ( A c –O, ) 3 r with r c 2 r r ), the r A )/ )/ Si– ( c 0.50 e/ r c c r II A r r r ( 2 ( ( 1.0 e/ ¼ r 0.74 e/ ; ) G G r 3 0.6 e/ r ( ¼ 1.62 ) and A > r r ) ( r ¼ ) ( c r , r 5 r . If we discount ( 3 A ) values generated r . In short, as the 0.97 e/ –O) A r 3 ( , ) and the A 5 r c ¼ Si– 2 r ) ( A IV and the ratio 6.2 e/ r ( r r is located relatively close ( 3 0.6 e/ R r ¼ c A r ) r , 1.36 e/ to ( 5 15.8 e/ 1.0. On the other hand, when ) is much larger than the value r c ¼ A 2 A r ¼ 1.0 e/ < ) ( –O bond [75] since Pauling first r r ) ) r ( r –O bonded interactions) and rela- c ( r r ( r 1.96 < 2 –O bonded interactions were each , 26.0 e/ r ) is relatively large ( 5 ) –O, c )/ ¼ r c ) ratios for these bonded interactions are r c ¼ A r c ( r ( Si– r ) ( r ( r r ( r G –O) VIII r )/ 2 c is located relatively far from the nodal surface 0.0, Si– –O, r 0.0, –O bonded interactions are all positive (except r ( , the interaction is classified as either an inter- 45.4 e/ –O bond in carbon monoxide which classifies as c 1 –O bonded interactions classify as shared except < Si– r VIII G > ´n [73] were careful to observe that an intermedi- ( ¼ Þ VI ) R c c ) r r r –O, ( ð ( ) values are greater than c The Bader and Essen [73] strategy employs the sign of There has been considerable debate regarding the elu- As displayed in Fig. 17, the r r r : 1.0 e r 2 2 2 3 Si– ( 0.0 and of the Laplacian,closed-shell interaction. then But if the interaction is classified as a both, then theinteraction. interaction In is general, classified the as further an the intermediate distance that associated with the closed-shell upper limit, from the surface and the smaller the value of values forreported the tocreasing Si– increase bondas progressively length follows: in and value coordination with number de- of Si ratio in classifying(covalent), a bonded intermediate interaction ortion. as either a A a closed-shell bondedr shared (ionic) interaction interac- isand the classified ratio as shared when > r that the A IV Bonded interactions and crystal chemistry: a review the Laplacian and the values of geometry optimized bondfrom lengths decrease progressively greater than 1.0, thenlifies each as of an theof interaction bonded the of interactions C– intermediate qua- character. The bulk for the C– intermediate in character. sive character of the Si– asserted onthat the it basis isthe of a literature, bond electronegativity several of workers considerations dominately intermediate consider closed the character. shell bond As ionicminately to [95, reported shared be 177–180], in covalent pre- othersate [181, predo- in 182] character and [6,properties others 137]. of intermedi- In the an electronof examination density of silicate distributions the molecules for physical [183], a the variety calculated mediate or a closed shell ionic interaction. If ate and particularlyas a a closed-shell result interactiontrated of is within the stabilized the atomicits electron basins proposal, of density Bader the beingBader-Essen bonded [2] atoms. rules locally and Since satisfactorily concen- lateractions classify workers for the have a found bondeddiatomic hydride, wide that inter- oxide, range sulfide,cules and fluoride together and relatively with nitride afirst large mole- large and number number second of ofsulfates. row molecules atoms involving and silicates, phosphates and greater the closed-shelland character Esse of the interaction. Bader for the M– for the bulk of the C– tively large, particularlymore for electronegative the atoms interactionsr involving like the Si, P and S. Also their to the nodal surface or r This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. –O b et al. ratio (dis- Þ c r Cl with large ð materials with ) ratio is less –O) and coor- c r VI r n ), the shorter the ( )/ ) to the local electro- c c (M– r ). Na r b 8 r ( R b ( )/ VI X c r n G r )/ ( c and ( M r G 1 ( G )e c r ) and the greater the shared ( c r r ( ) are given in ( )/ a c r r ( G. V. Gibbs, R. T. Downs, D. F. Cox G –O bonded interactions for a relatively large num- N with large covalent gaps were partitioned uniquely IV With these results, the decrease of B cal bond for a large number of character of atrends, bonded the interaction. anticipation On that the the basis of these IV than 1.0 forshell interactions shared does interactions not and appear greater to for hold closed- for the M– played in Fig.shared 24a) bond conforms character,bonded with interaction, the a the greater trendinteraction. greater the As of the such, shared ratio increasing actions the character for organization in of of a the Fig.suggests the 24a given bonded that into inter- the largelyis local connected disjoint kinetic to energy linearlyof density the domains per degree a electron towhole, given the which greater the bonded the shared value interactions character of is developed. On the into a domain containingnated structures atoms with only whereas four materials coordi- like ionic gaps were partitioneding into structures a with disjoint onlysis domain six-coordinated of contain- atoms. this On unique[187] the partitioning ba- that of structures,open covalent it structures was interactions with asserted tions well-directed favor and shorter lower the bonded coordinationactions interac- formation numbers favor more and of close thatted, packed ionic longer structures inter- with bondednumbers. less More interactions direc- recently, with thedecreasing coordination decrease larger number in coordination wasthe bond firmly length M– established with for ber of oxidesShannon by [30]. Shannon Further,strated and with Shankar Prewitt hardness [29] andplots sum and and Parr that later electronegativity Coulombic [188]coordinated by difference effects demon- crystal dominatedominate structures for for three six and and and four that coordinated eight structures. covalent effects bond length, the smallerthe the larger value of the coordination value number, of respect to spectroscopically definedergy covalent band and gaps. ionic On en- the basis of the gaps, materials like dination number with the increasing –O bonded interactions used in ( to a 0.3 A both de- . In a re- 1.29 c A ) the local kinetic energy density per electron, Cl, with six 0.02 to D a VI 1.12 Na –O) N, were asserted and . The symbols for the M– VI IV C– 1 ratio increases from i B III Na, Mg, Ca, Be, etc.; , of bonded pairs of ( h Þ IV )e h c c R c r ) ratio with decreasing ¼ r D –O) values than those popu- ð ) increases from 0.244 e/ c ( r r to r r ( ( M C– –Co) decreases systemati- both increase in value, the i I )/ )/ r r c ( h c A i r h , )/ r ( R h ( c (Co– H h r 1,3: A –O) with respect to ( G ( ) decreases from R c 1.39 ¼ G and r (M– , they found that the bonded inter- ( j R and as c , for the valence shell electrons and , c H i i and to ( D 1 c D h j h to 2.168 D A X –O) i and and A 3 C– M i IV A 1.16 h ( h R to 1.19 e 1 Scatter diagram of –O) –O) and the coordination number is consistent with Cl, O, S, C, etc.) solid state materials that exhibit the to 0.530 e/ C– In a highly cited follow-up paper employing dispersion The increase in the II ¼ 3 ( (M– dependence of aquantum crystal numbers, structure on the average principal X A nic energy density per electrons, R to consist ofclose-packed directed structures shared like bonded rock interactions salt, whereas cent study ofLasi Co-Co [185] metal found bonded that interactions, the Gatti and sively from Fig. 24. 28 theory, Phillips [187] examined the character of the chemi- coordinated atoms were asserteddirected to closed-shell consist bonded interactions. of more poorly the electronegativity differences, atoms [3]. They asserted that as cally from 2.691 R the famous Mooser-Pearsonnormal [186] valence separation diagrams for 0.35 e au, suggesting that themore trend general displayed by andtions. Fig. perhaps 24a may holds be for metal-metal interac- crease, the atomicteraction orbitals, become AOs, comprising betterted a developed whereas and bonded when more in- AOs highly become direc- moreteractions poorly developed being withdiagrams more the of poorly bonded in- actions directed. for Preparing a largelargely scatter number of disjoint crystals domains,with were partitioned the each into same populatedmore coordination open with structures numbers. structures withmore The four directed coordinated domains, bonded atoms with atoms with interactions, the with were smaller found to involve lated with structuresthe with six more coordinated opennation atoms. structures As numbers with such, like atoms with boron smaller nitride, coordi- This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. þ 29 –O ) –O, c r ( ) is the V c r ), then it ( –O inter- c ¼ G r –O, C– ) ( ), the more ) is a more ) decreasing ) is positive, c c c c c r G ). In contrast, r r r r ( c ( ( ( ( r H ( H H H H –O and Mg– r . As demonstrated –O, B– c r and ) actually increasing c j –O, and Mg– r ) (au) where ) ( c c r ) are not considered. As r ( H –O, Na– ( c ) is the local potential energy V r c ) such that V ( r j c ( þ r is greater than G –O bonded interactions dis- –O interactions with negative V ( ) , the greater the dominance of ) becomes progressively more > c –O, Na– c j –O) with respect to the local electro- c r G ) r ) is determined by the electron r ( ) > c c ( c r G r (M– r j ( ( H ( ) R ¼ c V H j G r ) ( c r V ( j H –O and S– ) values for the Be– c > r ( ) [173]. Cremer and Kraka [41] asserted that is larger than c ) with the local kinetic energy density domi- . As such, the Li– H r c c c j ( r r r ) –O, P– ( c G r r ) values for the Li– ( c Scatter diagram of V r j ( ) values classify as shared interactions with the shared ) is negative, then a bonded interaction is shared. On H c c –O, Si– r r ( ( density. local kinetic energy density and density itself, andinteractions as that such, resultdensity it from at is an negative accumulation for of all electron bonded G actions are each positive with actions inasmuch as theinterval local 2 energy density values in the appropriate measure ofthe bond Laplacian character. Unlike whichpression, the is sign the determined of sign by of the local virial ex- above, the more negative the value of indicates that thethe accumulation internuclear region of has the a stabilizing electron impact on density the in struc- negative (Fig. 26). If nic energy density, Fig. 26. bonded interactions classifytions as while the closed-shell remainingH ionic M– interac- the nating at Al– for each bondedand interaction coordination number with and decreasing increasing bond length shared the bondedtion of interaction, electron the density greater at the accumula- local potentialgreater energy the over stabilityabove, the of the kinetic the interaction energy [41]. and As the observed with decreasing bondcreasing length, coordination number and in- when observed above, the classificationother is factors based in on additionmer a to and number the of Kraka sign of argued the that Laplacian. the Cre- sign of character increasing as the other hand, when a bonded interaction isor a no closed electron shell density interaction accumulated with at little played in Fig. 26 are each negative with ¼ ¼ c c ) ra- 2.59; c D D r ( ¼ ) [172] r c c )/ r ) ratio for –O, ( c c D –O, r r r ( ( )/ r H c / ) ratio. In con- r Þ c ( c r ) ratios for the –O, r ( 1.07. As the ratio c H –O) and the elec- ð r 1.56; P– r ). With few excep- ( G c ¼ )/ r 2.03; Al– r c ¼ (M– ( c r )/ –O bonded interaction ( R c c r ¼ D –O bonded interactions r )/ H ( D c c oxyanion. In other words, r G D 2.74; Li– ( 6 –O, H ¼ –O, c oxyanion is greater than that for –O, D 4 –O and P– ) with respect to the c r 1.69; B– ( ) ratio for the Si– ), for the bonded interactions used to construct –O, r c c 1.02 and C– r r ¼ )/ –O, S– ( ( c ¼ r c 2.32; Be– r r / )/ ( c c D Þ ¼ r ), with respect to the local electronic energy density c G ( D c r r c ) progressively decreases with decreasing bond ð H ( decreases from right to left in the figure in the c ) for a given bonded interaction increases with r D G r –O, Al– c –O, ( )/ c r –O, c Scatter diagram of the local kinetic energy density per elec- ( r H D ( r G )/ –O, c As In formulating the underpinning for their strategy for r –O, B– ( order [189]: Na– (Fig. 24b). The scatteractions diagram organizes into the roughly bondedfor inter- the disjoint more linear electropositivegram M domains, of atoms the particularly [184]. A scatter dia- decreasing bondshared length character andalong for each coordination trend each fromas number, bottom bonded to the the top interactionbond in coordination length, the increases figure. the numberbonded Further, local interactions decreases kinetic (comprising a energy withis coordinated per indicated polyhedron) decreasing electron to for increase. As the such, the trast, the a silicate tetrahedral SiO Fig. 17. length for sharedconnection and was examined intermediate between interactions, a similar tronic energy density per electron, per electron, C– Fig. 25. tron, interactions considered inverse appears this to be study true. but rather the con- Bonded interactions and crystal chemistry: a review classifying bondedargued interactions, that Cremer the signder and and of Kraka Essen the [73] Laplacian [41] is as deficient employed in by classifying Ba- bonded inter- the smaller thethe coordination greater number the for locala a kinetic given energy given bonded density interaction. M per atom, electron for 2.00; Si– 1.36; S– tions, G tio organizes the bondeddomains interactions into (Fig. the 25) largely linear where the is negatively correlated with a larger silicate octahedral SiO Mg– are positively correlated with the This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) c to r ( et al. G ) and ) ratio c c = r r j A ( ( ) c ) less than G G r c = ( r j = ( ) j ) ratio greater V c ) 1.34 c –O data points j G r –O bond set c r ( = ( r j V ¼ ( ) j G c V = r –O bonded interac- j j ( ) V c j –O bond data sets r –O) . ( 3 –O bonded interac- V B– j –O interactions as the III ). As asserted by Espinosa ( c r R ( –O, S– r ), for the M– )/ c c r r ( ( –O) and coordination number G H –O, P– = j –O closed-shell bonded interac- (M– ) –O and S– c 0.8) to the C– R r G. V. Gibbs, R. T. Downs, D. F. Cox ( V j . Similar correlations exist for the –O, Si– , and magnesite, MgCO A 3 –O, P– ) is a direct measure of the shared charac- c r ( 1.50 G –O and Mg– ) ranks each of the data sets as expected from ) ratio and –O, Al– c c –O bonded interactions in agreement with Pau- ¼ = 2.45) in calcite and magnesite. Clearly, –O, Si– j r r ) ( ( c G G r Indicator ratio, ( –O) = = j j V ) ) j –O) increases from –O) and coordination number for the B– B– c c . [172], a bonded interaction with a ratio Rooted in the Bader-Essen [73] and Cremer and Kraka r r IV –O, Mg– –O, Al– ( ( (B– (B– ( V V [184]. If thecreases shared character with ofnumber, decreasing a bonded then bondj interaction length in- the and positive coordination correlation between the of data, withthe the ratio coordination increasingR in number value 1.51 increases to 1.63 from as 3 to 4 and C– between 1.0 and 2.0are is for intermediate calcite, interaction. CaCO The C– 1.0 is a close-shell interaction, one with a ter of afor given set the of Na–tions, bonded a interactions negative [184]. correlation However, exists between R and bond length in agreement with the[41] assertion. classification recipies,fies Espinosa’s the [172] M– recipeling’s classi- electronegativitywarned difference by Gatti recipe. [69],ing one conclusions should However, about bebased the solely as cautious on about nature the draw- ated of properties of at a the a bonded electron single density interaction bond evalu- critical point, however representative Fig. 27. R experimental bond lengthseach and increases. the Astory coordination such, for numbers the classifying a ratioit given does appears set not to ofin in be bonded character several satisfac- interactions, of cases, but ple, the appear a individual to positive bonds reflect correlation in the exists the between changes set. For exam- electronegativity considerations. The ratioB– increases for the observed for thethat bond data sets contradicts the assertion than 2.0 is a shared interactions and one with a et al bonded interactions ( tions for the mineralsto in the used to bond construct degree Fig. 17 parameter, plotted with respect j tions ( , ) ) j c c 0, þ ) r r c –O –O ) ( ( r c ¼ ( r r G 2 ) ( V = c j j r r G –O and ) ( c –O inter- r –N triple . [67] re- H ( V ) rather than j c 0, 2 et al r 1, shared when ( ¼ G ) < = c j r . [190] also found ) ) 2 each hold. With –O bonded interac- ( c c r r r ( ( ¼ 2 ) values calculated for ) ratio, the Ca– c et al ) V ) ratio that the K– c G ) is larger than r r j ). Nonetheless, the ratio c c r = ( c c r ( r from the set of Na– j r r ( ( r ) ( ( –O bonded interactions as G c ) ratio where G G r c = c G H ( j = = r j ) j D ( V ) ) c j c c r r r ( r )/ ( ( )or c V c j V V r –O match those calculated with a r j j ( ( H r , a situation where there is no compo- 2 –M bonded interactions. r i.e. 0 [73]. For the case where 2 and intermediate when the ratio falls ) is positive as expected for a nonbonded –C bond in the gas phase molecule 0 and when c 1 and < 0 [41] and that it is a shared interaction r ( > ¼ ) –O bonded interaction comprising the ¼ c –F bonds in the closed shell region and the > ) H ) ratio was calculated and plotted in Fig. 27 ) ) r were observed to be the same as that observed c ) ( –O and Mg– c c c r –C bonded interactions in a hard mineral like c r r r r 3 ( ( ( ( r ) for closed shell and shared interactions. It was 2 ( V c 0. For these two constraints, the two equalities G G G r r H ( –Si bonded interactions displaying more closed- = = = þ j j j ¼ –CH ) ) ) H ) . [172], is based on the ratio ) –O bonded interactions in diopside classify as closed- c c c c c On the basis of the The third classification, ably proposed by Espinosa For the minerals used to construct Fig. 17, the r r r –O, Na– C– r –O– –O interactions [103]. In an experimental study of tran- r –C single bond for ethane in the shared region, demon- ( ( ( 3 ( ( V V V between 1 and 2. procrystal representation of[140], the as electron asserted densityno by indicating Bone electron andpoint density Bader when is [173], that accumulated little at or the bond critical j situation. Like theCremer Bader-Essen and classification Krakareasonable [41] (1994), success classification the hasbonded in been the interactions used with classification ranging of from a LiF variety to of C– bonds, including M– et al is based on the constraints assumed to exist for increases with decreasing and ture. If, on the other hand, 30 assumed that anwhen interaction is a closed-shell interaction G directly on either for the C– diamond [191]. j tions were found tosubstantial be intermediate component in ofbridging character but closed-shell with character a Si– with the on the basis of the Si– shell characterSi– than thesition metal nonbridging compounds, typically Bianchi shorter shell interactions as expected. The Si– Mg– V these equalities,closed-shell a when bonded the ratio interaction is asserted to be with respect to the bonded interactions classifyactions as as closed-shell, intermediate Mn– shared and interactions. Cl– In addition, Gervasio then the accumulationbonded situation is ( destabilizing,nent typical of of covalent bonding). a The non- Li– when j ported that a varietyging of between metal-metal 1.0 bondsbetween with and Na– ratios 2.0 ran- C– lie instrating the that intermediate metal-metal bonded region ate interactions rather are than intermedi- sharedthe interactions. metal-metal They bonded also interactionslecular observed complexes in that have organometallic the mo- density same properties physical as and those localnot for energy considered bulk surprising metals. inasmuchties This as result the was of physicalH proper- the C– This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , ) 3 r 31 ( S and 2 r octa- D 2 energy 6 low spin 2g t octahedron 0 . [193], the ) 6 g in height at e ( , millerite, NiS et al 2 6 3 ) A 2g t –S bonded interac- , pyrite, FeS ) values are slightly 4 , cubanite, CuFe r 4 S ( 3 S –S bonded interactions 3 G . [130, 137, 155] and later and 1.6 e/ –S bond vectors, it was not [198] and for the Fe and Cu et al 3 in pyrite by evaluating the ) being roughly 10% larger than 2 A –M and S– r S ( þ 3 2 levels. Using the parameters gener- G , greigite, Fe 2 g for the large values. A regression line are seen to be highly correlated. The e i –S, M– i ) ) with c c r and r ( i was calculated. These values are compared ( ) g G r G ( h a Þi h c G r h from Fe, as expected for a ( ð maxima along the Fe– G . Overall, as asserted by Espinosa h A i Þ ) ) and . [197], made an important advance in a careful study r ) vs. c r ð Using single crystal X-ray diffraction data, Stevens ( r r 0.6 ( r ( -orbitals of the Fe atom. The maps revealed eight local G through the dataG points yields a nearly perfect correlation agreement is comparablelarger but than the configuration, directed toward the faces of the FeS maps and establishingin a the connectiond distribution between the and features maxima the ligand between field 1.2 splitting e/ of the sulfides trolite, FeS, smythite, Fe Physical and local energyfor density the properties electron densityfor distributions transition metal sulfides Sulfides comprise an importantbit class a of range minerals of that M– exhi- and structure typestant [194] electronic, in magnetic concert andto with catalytic understand a properties. and host Ifserved of exploit we above, impor- their are an manifold understandingthe uses, of then bonded the as interactions connectiondensity ob- and between distribution the at properties thethe of atomic past the level is electron pended two required. to decades, During structures, advance considerable their our effort stabilities understandingsemi-empirical has and and of been physical more sulfidetions ex- recently properties, in crystal first-principles the using As generation calcula- documented of byby the Gibbs Vaughan electronic and Rosso structurescalculations [196], [195]. to it is generate nowtures bond possible lengths together to and with useserved angles the their within of a properties struc- fewstanding that percent, of thereby sulfide match crystal enhancing those chemistry. our ob- under- et al of the spin state of Fe and the gassion, theory approximation and the virial expres- chalcopyrite, CuFeS ated in a multipolesuccessfully refinement, calculate they the werethe not electric magnitude only field able ofbauer to gradient the spectrum. but quadrupole-splittingD On also of the the basis Moss- of the absence of local levels into h with thoseG calculated for the olivines in Fig. 28 where resulted in a splitting of the three degenerate hedron. A slight trigonal distortion of the FeS and heazlewoodite, Ni clear at thetions time in why pyrite thebonds. are low substantially More spin shorter Fe– recently,calculated than electron for the density the high distributions Ni spin sulfides were vaesite, NiS electron gas approximationthe performs local well energywith in properties a for reproducing relatively accurate a electron material density when distribution. generated 4 6 5/ 0, = 1 r –O SiO ¼ 2 2/3 ) ) and ) . With c c 2 i r and the r ) and the ( p ( [161] (red Þþ c r r r i r 4 ( Þ ð ) = D c c r V i r r / ) ( ð 2 SiO c 2 ), estimates of r r )] when evaluated G )] c ( c –F closed shell h r r r ( . [184] to repro- ( ( G iþ h (3/10) (3 h r ) ) can be estimated r r 2 c c 2 ) used to generated r r r r c ( ( [ et al [ r i¼ r l 6 V ) ( G = c h 1 ) calculated for the M– r r c 2 2 ( r þ þ ( r G ) ¼ –O and the (Mg, Fe, Mn, G h c ) and ) r ) c ( r [164] (orange spheres) and the sili- c r ( r ( ) can be determined. Using the 4 5/3 ( c . [193] concluded that the elec- r r 5/3 r r ) and ( 2 r c SiO 2/3 2 r H ) r ([162] (black spheres), tephroite, Mn ( 2 et al 4 p 4 r = 2/3 1 ) 2 )]. With the value of c SiO p 2 r ( r 2 (3/10) (3 r ) were found by Gibbs [ c ) and hence i¼ r 6 ) c ( c = (3/10) (3 r values generated for the model calculated electron 1 r r ( The local kinetic energy = Þ )] [192]. At a bond critical point where G c i r h þ ) i¼ r ( c ) ), V( ð –O bonded interactions for the four olivines [165], ), r c r r ) c r 2 ( ( c r r 2 ( r ( H G ( r spheres), fayalite, Fe bonded interactions for the olivines forsterite, Mg the theoretical values for h at a bond critical point. r with the expression Fig. 28. [163] (blue spheres) and Co [ To our knowledge, thereing is no the direct local strategydirectly kinetic for from determin- and athe potential local model energy kinetic electron densitygradient energy density density values corrected distribution. canh be But, estimated electron with the gas theory approximation, Estimates of the localand kinetic potential energies fromelectron experimental density distributions it may be. Forfication a of more chemical comprehensiveurged review interactions of to in the consult crystals,crystals classi- [69]. Gatti’s the reader review is on chemical bonding in Bonded interactions and crystal chemistry: a review 3 calculated values of ca polymorphs coesiteagainst (black stars) the and gradientG stishovite corrected (blue stars) electron plotted gas theory approximation of local virial expression, the value of G density distribution for the Si– Co)– the plots in Fig. 17, the values of duce Figs. 25afound and between the 26.local calculated With energy and the density estimatedinteractions, comparable values properties Espinosa agreement of for the tron 37 gas H– theory approximationmate provides of a satisfactory the esti- local properties. With the the approximation reduces to This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. –Ni –Ni –Ni et al. ). A ( R ) calculated for b –S bond paths –Ni bond paths –Ni separations )( c –Ni bonded interac- r ( r 2 –Ni bond paths was ) values for vaesite, r c r ( , each Ni atom is con- r The bond critical properties 2 A ) and –S and Ni– a r –S bonded interactions and , each Ni atom is connected )( c –Ni bonded interactions for bulk Ni –Fe bonded interactions [201]. A r ( the Ni– Fig. 29. r tions for the Niite sulfides vaesite, and miller- Ni– heazlewooditemetal and plotted forperimental with bond respect lengths, the to the ex- ) and G. V. Gibbs, R. T. Downs, D. F. Cox c ). In millerite, with a nearest neigh- r ( b A r –Ni bond vectors, as observed for mill- –S bonded interactions (Fig. 29). As ob- –Ni bond paths were found in vaesite, –S bond lengths in Fig. 29, increase linearly –Ni bond paths were found to connect the ) are substantially longer than that observed for –Ni separation of 2.53 –S bonded interaction in heazlewoodite. This would A The calculated –Ni bond paths that extend uninterrupted throughout will also belinked of by bond interest pathsstructure to in is see support stabilized whether by of Fe– the the conclusion Fe that atoms the are millerite and heazlewoodite, plottedperimental with Ni– regard towith the decreasing ex- experimentaldeveloped bond Ni– length.nearest Further, neighbor well- Niwoodite. atoms No in Ni–probably both because millerite the(4.02 nearest and neighbor heazle- Ni– bor Ni– bulk Ni metal (2.49 nected by bonddiscrete paths rings to of two threeIn adjacent contrast, Ni in atoms Ni heazlewoodite connecteddistances atoms, with by of forming nearest bond 2.50 neighbor paths. by Ni– and 2.54 fourNi– bond pathsthe forming structure. a Afound similar for continuous bulk array network Nito of metal twelve of Ni– nearest where neighbor eachthat Ni Ni extends atoms atom uninterrupted by is throughoutthe Ni– the connected metal. bond Moreover, bonded critical interactions point formilar bulk properties Ni to calculated are thosebut not for calculated collectively only they Ni– for strikingly followdefined si- millerite along by projections and the of Ni– served heazlewoodite the above, trends themetal bond critical bonded pointcomplexes properties interactions are for metal- also observedfor virtually the for the bondedThe same organometallic interactions accumulation as of comprising the thosebond bulk electron observed paths metals density together along [67]. are with the indicated Ni– that to alongstabilizing stabilize the the impact S– structures ofthe beyond S– the that of Ni– be the the case if,lates at along equilibrium, the the Ni–erite electron and density heazlewoodite, accumu- becausemore it than stabilizes if thedistribution structures distributed reached differently. at As equilibriumyielding is such, the the the lowest bestas electron energy compromise, such, for the therole presence system of in as bond a stabilizingdelocalization paths whole a of is and structure. the indicated electrons The to in play high heazlewoodite a conductivity is and consist- in –S –S A were . The Fe– 4 A VI S 3 to 2.40 ) ratio classi- c r to 2.15 A ( a –Ni bond critical G A = , the critical value j ) c A r to 2.31 ( V j A and greigite, Fe –S bonded interactions and 2 Fe– VI –Fe separation along the unit cell , respectively, as the coordination –Ni interactions. . Calculations for troilite, cubanite, orbitals overlap extensively [200], it þ –S bonded interactions are connected þ 2 2g whereas pyrite and marcasite both con- t –S bonded interaction for pyrite is great- Fe to 4 [199]. In these studies, the local properties þ VI 2 Fe– 2 þ VI Fe –S and Ni– VI ,3 in troilite is less than 3.0 , chalcopyrite, CuFeS þ 4 c S 2 , 2.5 The Fe-sulfides troilite, pyrite, and marcasite were cho- The three Ni sulfides are of interest because they cover –S bond lengths decrease from 2.45 þ heazlewoodite, millerite andmatter of vaesite, interest respectively, to it see how is the a whether the shared character of the low spin er than that for high spin sen for thespin state second study because troilite contains high fies the Ni– tain low spin marcasite, FeS 32 bonded interactions isinteractions. greater As than the that Fe– for the high spin vector also undertaken becausethe the Fe atoms nominal2 are oxidation indicated states to of increase in the series from CuFe point properties were expectedand to the be bulk similarof in metal. the the Further, sulfides Ni as atoms the increase coordination from numbers 4 to 5 to 6 as the Ni– a remarkable rangeinsulator, of millerite electrical isis properties. a a Vaesite semiconductor conductor. islated and Electron an for heazlewoodite density the three distributionsbetween to were the establish calcu- local whetherthe a electrical properties connection properties. of exists Asatoms the the separations in electron betweensame the density millerite as Ni and and those in heazlewoodite bulk are Ni metal, virtually the the Ni– for the electronthe density same distributions first were principlesculate generated strategies using the employed properties abovefor of to a variety cal- the ofparison, electron oxides experimental and density physical silicates.mined distributions properties For for purposes were of heazlewoodite alsoexperimental com- in deter- electron a densityhigh multipole resolution distribution modeling single generated crystal of X-ray with the diffraction data. bond lengths increase from 2.27 below which the calculations were alsoproperties of undertaken the toto Fe– the see spin howordination state, numbers the the of nominal local theto oxidation Fe see states atoms. whether and It thethe was the accumulation low also of co- spin of the interest electron density for number of theFe– Fe atoms decreases from 6 to 4 and the This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) þ b c þ 33 r ( 2.5 –S) r (Fe– –S, the (Fe) for ) values R b c for Fe r r ( –S bonded 3 – . Given that r 5 þ A 2 ) is typically A c r –Ni interactions Fe ) values for the ( c –S bonded inter- – VI r r 2 in chalcopyrite to (S) of the S atoms ( þ 0.48 e/ ), poorer agreement b 2 0.5 e/ r ) for the Ni– c r r ) increases with the c r þ c r Fe 3 r ( ( VI r r 2 .As r 3 for Fe A ) results is poorer overall molecule was geometry-opti- ), –S bond length and the values c c troilite to 3 ), the bonded radius, r r 4 ( c ( A þ r r 2 r ( 2 r 0.35 e/ 2 in greigite. As there has been debate FeO r value for greigite, it was concluded 4 ) with two well-developed trends, one r þ Þ c 4 0.53 e/ c r r ), ( –O bonded interactions and the other for ð c r r ) values for the low spin –S bonded interactions are compared in r ( c r a r 0.35 for Fe ( for Fe r 3 A –S bonded interactions. The –S bonded interactions are plotted with respect to ) closely matched the experimental bond length and ) is the second derivative of ( –S interactions. c c r r ( ( As observed above, a set of high-resolution single crys- The experimental bond lengths for the high and low –Ni and S– –S bonded interactions and the other involving the low –S bond length in much the same way that r r 2 0.74 e/ 0.6, than that for the high spin Fig. 30a where theagreement agreement is comparable. of However, the the Ni– have a slightly greaterthe component Ni– of shared character than tal X-ray diffraction datational has aspherical-atom been multipole collectedof and refinement the a and conven- experimental modeling dertaken electron for a density syntheticof distribution heazlewoodite the was crystal. gradient un- Aeach field mapping of of the thefaithfully electron matched bond density by pathstion. a revealed in The path that experimental the in values the calculated of experimental distribution distribu- is highly correlated with (Fig. 30b) with observedexceeding values those for calculated the by Ni– about the Fe atoms and the bonded radius, for high spin Fe– fall along two separateFe– trends, one involvingspin the Fe– high spin in Figs. 32a through 32d, respectively. The mized with asulting relatively minimum robust energy basis Fe– of set [199]. As the re- also increases withatom the from nominal oxidation state of the Fe high spinlength and interactions the coordination increase number with of the decreasing Fe atom. bond correlates with Fe– in cubanite to the calculated interactions in pyrite and marcasite are substantially larger, that the oxidation state of the Fe atom is likely to be 4 [199]. The r about the oxidationgeometry state of of Fe in the greigite H [205, 206], the is to be expected. spin Fe– the value of actions in troilite, ) ratio in- c r –Ni bonded ( G = j ) c r –Ni separations of -type orbitals loca- –Ni bond paths in ( clusters are hopping p V 9 j –Ni bond paths. S 3 –Ni bond paths in miller- –Ni -type electrons in the 3-mem- ) val- d b )( and the coordination number of Ni c r ( A r 2 r –S and Ni– –Ni bond paths. Clearly, the accumulation –Ni interactions are intermediate in charac- to 2.27 ) and a clusters into a structure of NiS composition. rings that each comprises a closed circuit of –S, S– charge-transfer interactions [204]. It is asserted rings comprising the Ni A )( 3 9 3 c p have also been pictured as potential circuits for r A comparison of the ob- S ( 3 –O bonded interactions, the r A -S2 d The local energy density properties calculated for the In contrast, the stabilizing Ni– –S and Ni– –S interactions are indicated to be intermediate in char- –Ni bond paths of accumulated ED. However, the rings in tandem tolized adjacent on rings thetransport via coordinating of the S millerite is atoms.observed expected As for to such, heazlewoodite. bethe the However, not conductivity as electron as good ofsuch is as factors well a that as known, NiSthermal impurity scattering, crystal scattering, factors isThe grain that insulating also boundaries were properties ignored dependent and absence of in on vaesite of the are Ni– of study. consistent with the its electronheazlewoodite density and millerite alongtures in serves the excess to Ni– of those stabilize that the lack Ni– struc- ter between closed-shellthe M– and shared interactions. Unlike bond interactionsNi– for the Ni sulfides indicate that the interactions in millerite and heazlewoodite are indicated to decreases from 6Ni– to 5acter to with 4. ater. substantial According component On to of the the closed basis shell ratio, of charac- the a ratio of 1.35, the Ni– creases from 1.20from 2.40 to 1.26 as the bond length decreases bonded interactions inwith heazlewoodite those calculated. ues for Ni– Unlike the metallicwoodite, conductivity the electron ofNi3 transport bulk has Ni been and related heazle- to strong bered Ni Fig. 30. served ent with[202]. its The well-developed metallicwith and the conductivity presence robust canconnect of the band also Ni the atoms beconnected structure well-developed in bond connected bond a highly paths. pathscan branched The that be circuit end of picturedaccumulated inter- product as electron wired is with density acrystal, a ideally that crystal circuit suited that radiateing for of throughout dipyridylamide electron bond transport. the molecules paths2.45 Two with of Ni Ni– bear- ite are restricted tobered a Ni periodic array of discrete three mem- Bonded interactions and crystal chemistry: a review electron transport [203]. are connected by coordinateding S Ni atoms that link the result- Ni– that electron hoppingand occurs S between nearest-neighbor atoms Ni where the This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) ) ), c c a r r –S ( ( –O) et al. G H ). The = ), with j d (Fe– )in( ) with a b c c R r ), and the r ( ( c –S) for the r r ), the local V A ( j c r V )in( ) the local ki- ( (Fe– –Fe distance c (S), in ( b r b G R ( r orbital directed r 2 ) ratio orders the –Fe separation in 2g r c t r ) and ( ( c . As no bond paths r H  ( ) and with respect to the . The Fe– c = r c j –S bonded interactions )/ ) ) The indicator –S interactions are clas- c The bond critical point prop- c a r r ( ( ( H V j –S bond lengths, (Fe), in ( b Fe-sulfides identified in (a). is plotted with respect to bonded radius of S, properties for thepear low spin to trends tracking ap- along the high the spin line trends. parallel- respect to ther bonded radius of Fe, with respect to Fig. 31. erties calculatedspin for highplotted Fe– with and respectFe– low to the observed netic energy density, potential energy density, plotted against themeter bond degree para- plotted with respect to the the Fe– bond lengths observedfides. for See the Fig. 31a Fethe for symbols. sul- the definition of Fig. 32. local electronic energy density, octahedra share faces [208]. This G. V. Gibbs, R. T. Downs, D. F. Cox , the Fe occupied b  6 b d –S angle of 82.6 –Fe bond paths were found for troilite –Fe– is substantially longer than that in bulk Fe ). A A Figure 32a shows that the –S bonded interactions in–S the same bonded way as interactions, it orders with the interactions involving –S bonded interactions, the Fe– sified according to the ratio as bonds of intermediate char- between the edge sharing octahedra is 3.38 shared edge S– Fe– Ni– shorter bond lengths,metal atoms smaller with coordinationplay larger numbers oxidation more and statesNi– shared indicated to covalent dis- character. Further, like the were found between theangle Fe is atoms less and than as 90 the edge sharing dra parallel to the cell edge vector between the Febonded atoms such in that the the[196]. chain Fe is Also, atoms indicated no mutuallywhere to Fe– repel be adjacent one anti- another FeS may not betrolite, surprising 2.99 given thatmetal the (2.50 Fe– ) ), c c r –S – r ( ( þ r r 2 Fe VI octahedra in a 6 c a –S bonded inter- dimer in pyrite and 2 ) values calculated for the molecules is c r 6.5 GPa [207]. Unlike the FeS ( r synchrotron X-ray emission study has demon- 2 smaller than that calculated for greigite. The ), the bonded radii of high Fe and the S mono- b c r 5 r K ( A r ) is substantially larger for the low spin 2 c r r ( r 2 0.5 e/ strated a reversible high-lowin spin troilite transition at of the Fe atom r and mer each decreasesber with and bond the oxidationthat length, state a coordination of Fe num- the Fe atom. It is of interest low spin bonded interactions (Fig. 31b). Like 34 bond radii fortrends, Fe again and one S involving atoms high each spin scatter Fe– along parallel actions and theIn other addition, involving the theand low low the radius spin spin of interactions. bonded the S radius atom of in the the S Fe atom marcasite are substantiallyatom smaller and than the the S high monomer spin in Fe troilite. However, like pyrite, which onlysite share share corners, edges the and octahedra form in chains marca- of edge sharing octahe- bonded interactions thanever, the the high spin interactions. How- This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ) r 35 ( –O –O –O, L ) and Si– c . r IV ( A r –O, Si– –O, Na– ) values. In c r 1.25 ( H –O), the coordi- –O bonded inter- ) ratio, c to –O, Al– decrease. With the ) when bonded to r ( (M– –O, Be– A c A r R century, Bader and his D )/ c r th –Si angles are the most ( –O, C– 1.50 0.95 G –O– , when bonded to electronega- ) and negative A c r ( –O and several C– r 2 0.65 ) not only predict that the O atoms ) maps for the aluminosilicates also r r r ( ( ) values for the Li– L ), and experimental and theoretical h c r r ( ( h H –O bonded interactions examined in the study ) maps generated for the common rock forming –O bonded interactions of 1.0 and 0.67, respec- –O bonded interactions are positive and accord- r ( ) each increases in value and –O bond with the bond valences for the Si– r in stishovite and perovskite. Indeed, experimental c r VI D ( þ The local maxima displayed by the electron localiza- The shared character of the B– During the latter part of the 20 r 3 2 O, ranging in one case from –O and S– b and minerals were foundbond to and display such lonenumber, covalent electron size features pairVSEPR and as domains location modellone that with for pair agree those closed-shell featurestential in predicted were molecules. the chemical found by Further, tovance activity, the represent the the information regions understanding thatdefects of of in po- served minerals the toAl and role the ad- played incorporation by of elements hydroxyl like crust, structures thattures closely determined in matched highthe experimental pressure experiments. structures struc- Indiopside for addition, and rock talc,likewise forming generated found minerals for towithin ambient like a agree conditions few tremolite, with percent. were the experimentaltion structures function, P– is indicated to increase as the colleagues forged a powerfulory and for abstracting widely important usedsical information natural properties defining the- the oftron phy- the density bonded distributions interactionsGatti for [69] from for molecules the anoretical and elec- excellent and crystals review). experimental In (see tions model a generated electron survey for of density athe the number distribu- properties the- of not silicates,Pauling’s it only second was provide postulate found in adistribution, that terms basis but of they for the alsopirical understanding electron reproduced bond density the lengththe well-known – em- Si– bond valence trend established for r and Mg– ingly they classify as closed shell ionic interactions with tively, matching the experimentalthe and electron calculated density valuesactly. at Rather of the than bond remainingO critical fixed, atom points the yielded almostdensity bonded by ex- radii distributions of theoretical rangeradius an and of in experimental the value atom, electron between the atomic mappings of the involved in thereactive narrowest chemically, Si– butstudy also to be are bondedrecent those to study found H in of inserved a as a specimen a key with FTIR for Si interpreting defects. the A FTIR spectra [212]. contrast, the and actions, the interactionstions with classify positive as intermediate interac- nation number of the M atom and electropositive Na [54].radius Rather of an than individuallarly remaining O atom when fixed, is bonded the highlyplays to polarized, a particu- several different differentr M value atoms along and each dis- of the bond vectors, tive N and the ionic radius ( exception of a single S– ) in- c r ( G local energy j ) c r ( V –S bonds as sug- – –S bonded interac- j – þ þ 2 2 Fe Fe VI VI ) and –S data also scatter along c r ( G ) both decrease with decreasing c r – bond number trends displayed ( –O bonded interactions, H –S bond length. As observed in Fig. 32, ) and –S bonded interactions are displayed in c r ( V With the development of first-principles strategies density values. experimental Fe– the high spinparallel and trends the with low larger Fe– A number of theknow fundamentals them of today, were crystalthe established chemistry, structure as early determination we lastmaterials of century a (many with relatively of largeof them number atomic, minerals), of ionic thesal and derivation crystal of radii of athe together set sets with factors of the that postulatesstructures. propo- govern by Of the Pauling the [22] relative postulates,way for the stability second that establishing of revolutionizedminerals alternative the mineralogists in view termsassertion of the that theirbonded crystal the interactions bonded chemistry reaching sum interactionstends a of of with to given the anion satisfy theSets in the bond of a bondingschmidt structure strengths empirical requirements [15], for radii ofmuch Pauling later the derived the [19], by anion. Shannon byserved Slater and to Prewitt Bragg [21] facilitate [29]large and the [17], and Shannon determination number others [30] Gold- and ofand and verification crystal testing of their structures, a derstanding bond not of lengths, the only but relativewith also reproducing stability adding the of to alternative constructionfound structures our wide of un- use structure inranging the field modeling from maps. of iontions. a They variety conductivity Relativistic also of to phenomena self-consistentmost trace valence field shell element radiiatoms electron distribu- of are density highly the radial correlatedoretical outer- maxima with underpinning the for for radii, the portant providing ionic advances a and were the- atomic alsobond made radii valence in [20]. the Im- andlength demonstration with that bond a order powermodel law the are expression bond similar connected to length with that – used bond to based on density functionalmentation theory of (cf. the [3])102], and theory CRYSTAL98 the [149] with imple- advances and software were TOPOND like made [156], VASPand in important [100– the the modeling physicalcrustal of pressures properties crystal but structures of115, at 128, minerals, mantle 155, not pressures 210,the as only silica 211]). polymorphs well For under were example,conditions (cf. generated the [95, but not structures only for for for ambient pressures encountered in the Earth’s Concluding remarks by hydrocarbon moleculesLater, the and strategy connecting intermetallicwas bond refined compounds. length and to itdes, bond was valence found fluorides, thatwithin sulfides the a bond few and lengths percent, in without nitrides oxi- resorting to can tables be of radii. reproduced acter. Also, the low spin state Bonded interactions and crystal chemistry: a review creases and tions incovalent pyrite and than marcasite high are spin indicated state to be more gested by Cottonties [209]. for The localFig. the 32b. energy Fe– density Like proper- the M– This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . 4 72 –Al et al. SiO 2 (1993) (1999) (1964) (1929) (1977) 41 31 68 51 62 Reviews in (1898) 433– 29 (1958) 552–579. (1969) 925–946. 43 25 (1928) 168–193. (1929) 291–314. (1969) 1528–1539. 69 25 (1927) 765–790. 54 49 (1976) 751–767. (1994) 481–510. G. V. Gibbs, R. T. Downs, D. F. Cox 32 50 (1926) 538–556. (1990) 633–643. Silicate Pyroxenes. Am. Mineral. (1883) 186–188. 63 46 . Z. Kristallogr. c 2 (Ed. P. H. Ribbe), pp. 275–381. Mineralogical So- 29 ) 2/ 3 (1920) 169–189. (1923) 1–25. C 40 38 (1929) 537–559. –O bond. Am. Mineral. Z. 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Crystals:1960. Cornell An University Introduction Press, to Ithaca, Modern N.Y. Ordered 710–720. Mineralogy ciety of America 1982. [8] Belokoneva, E. L.: Electron density and traditional structural [3] Parr, R. G.; Yang, W.:[4] Density-functional theory Bragg, of W. L.: atoms Atomic and Structure of[5] Minerals. Slater, Cornell University J. C.: Introduction[6] to Chemical Pauling, L.: Physics. The McGraw-Hill, Nature of the[7] Chemical Hawthorne, Bond. Cornell F. Univer- C.: Minerals, Mineralogy and Mineralogists; [9] Stillwell, C. W.: Crystal Chemistry. McGraw-Hill Book Com- [25] Warren, B. E.; Bragg, W. L.: The structure of diopside, [10] Evans, R. C.: An introduction[11] to Barlow, crystal W.: chemistry. XXVI. Cambridge Geometrische Untersuchung uber eine me- [12] Barlow, W.: Probable Nature of the Internal Symmetry of Crys- [13] Perutz, M.: How W. L. Bragg invented X-ray analysis. Acta [14] Bragg, W. L.: Atomic Arrangement in the Silicates. Transac- [26] Warren, B. 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B.: The structure of olivine (Mg, Fe) [32] Ribbe, P. H.; Prunier, A. R.:[33] Stereochemical Brown, Systematics G. of E.: Olivines and silicate spinels. In: ) c r –O ( H is from the c r –S bonded inter- –O bond, the most ) increases and as c r ( r –O and M– –O bond classifies as a bond of ) together with the values of r ( ), and the distance that r r 2 ( r r The National Science Foundation and the U.S. 2 ), the Si– c r r ( G = j ) c r ( Wesley Longman, New York 1970. tions, Oxford, UK 1990. V –O) decreases. The elusive Si– j ) increasing in value as c Finally, the topological properties generated for experi- r [1] Feynman, R. P.: The[2] Feynman Bader, Lectures R. in F. W.: Physics. Atoms in Addison Molecules. Oxford Science Publica- ( (M– and nodal surface of References Acknowledgements. Department ofgrants Energy are EAR-0609885CTP), thanked (NLR for and andgrant supporting GVG), DE-FG02–97ER14751 from this EAR-0609906 theEnergy (DFC). study (RTD U.S. Sciences, with KMR Engineering Department and from and of acknowledges Geosciences the Energy Division, a Environmental (DOE),the and support Molecular Office Pacific Sciences of Northwesttions Basic Laboratory were National performed (EMSL) Laboratory innational at scientific part (PNNL). user at facility TheBiological the sponsored EMSL and computa- by at the Environmental PNNL.telle U.S. Research. The for DOE’s Office EMSL the PNNL of isfully DOE is a under acknowledges operated Contract the byand DE-AC06-76RLO help 1830. Bat- the AK by financialForschung, grate- W. contract support Morgenroth, No. by 05 HASYLAB/DESY thanking KS1 the Bob PDA. Finally, Downs Bundesminister GVGversity for takes fuer of pleasure providing Bildung in Arizona supportmanuscript during und was for the written. his winter visit of to 2007 the where Uni- the bulk of the H 36 R abundant bonded interactionstudied in by the a earth’sclosed number crust, shell of has ionic workers been bymediate and several was in and concluded shared character covalentthe to by and Laplacian, be inter- others. However, on the basis of actions. The agreement indicateslations that coupled first with principles theeling calcu- theoretical and of experimentalprovide mod- electron an important density quantumstanding mechanical the distributions basis local for densitytions, are properties under- the of destined the crystalchemical bonded to chemistry interac- reactivity of andyears minerals the to at properties come. thein and Also, atomic the the local level development calculationssults, in and will but the interpretation they not of willthe only also experimental uses assist provide and re- a crystalcal deeper chemistry processes, understanding of enriching of minerals the field and of mineralogi- mineralogy. intermediate character inelectronegativity agreement classification. with It Pauling’s issical classic pertinent properties that of thethe phy- the local electron energy density density distributions and properties classify the M– mental model electron densitylution distributions with and high highdiffraction reso- energy data are synchrotroncalculated in single for comparative a crystal variety agreement X-ray of with M– those bonded interactions prettymous much classification in line oftronegativity with bonded Pauling’s interactions difference fa- basedPauling on considerations. classification, elec- thetron But physical density properties provide unlike a ofacter means the the of of elec- determiningsuch an how the factors individual char- aslength bonded bond and strength, interaction coordination thelocal changes number, energy local bond density with kinetic, distribution. potential and the electronic This article is protected by German copyright law. 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